Least energy nodal solutions of Hamiltonian elliptic systems with Neumann boundary conditions
LLeast energy nodal solutions of Hamiltonian elliptic systemswith Neumann boundary conditions
Alberto Salda˜na ∗ & Hugo Tavares †‡ November 9, 2018
Abstract
We study existence, regularity, and qualitative properties of solutions to the system − ∆ u = | v | q − v in Ω , − ∆ v = | u | p − u in Ω , ∂ ν u = ∂ ν v = 0 on ∂ Ω , with Ω ⊂ R N bounded; in this setting, all nontrivial solutions are sign changing. Our proofsuse a variational formulation in dual spaces, considering sublinear pq < pq > not radial functions; a key element in the proof is a new L t -norm-preservingtransformation, which combines a suitable flipping with a decreasing rearrangement. Thiscombination allows us to treat annular domains, sign-changing functions, and Neumannproblems, which are non-standard settings to use rearrangements and symmetrizations. Inparticular, we show that our transformation diminishes the (dual) energy and, as a conse-quence, radial l.e.s. are strictly monotone. We also study unique continuation propertiesand simplicity of zeros. Our theorems also apply to the scalar associated model, where ourapproach provides new results as well as alternative proofs of known facts. Keywords
Dual method, subcritical, Hamiltonian elliptic systems, flipping techniques, sym-metry breaking, unique continuation.
Let N ≥
1, Ω ⊂ R N be a smooth bounded domain, and consider the following Hamiltonianelliptic system with Neumann boundary conditions − ∆ u = | v | q − v in Ω , − ∆ v = | u | p − u in Ω , ∂ ν u = ∂ ν v = 0 on ∂ Ω , (1.1)where ν is the outer normal vector on ∂ Ω and we consider p, q > sublinear ( pq <
1) orthe superlinear ( pq >
1) cases satisfying a subcritical condition, that is, p, q > , pq (cid:54) = 1 , and 1 p + 1 + 1 q + 1 > N − N . (1.2) ∗ Institut f¨ur Analysis, Karlsruhe Institute for Technology, Englerstraße 2, 76131, Karlsruhe, Germany,[email protected] † CAMGSD, Departamento de Matem´atica, Instituto Superior T´ecnico, Universidade de Lisboa, Av. RoviscoPais, 1049-001 Lisboa, Portugal; [email protected] ‡ Departamento de Matem´atica, Faculdade de Ciˆencias da Universidade de Lisboa, Edif´ıcio C6, Piso 1, CampoGrande 1749-016 Lisboa, Portugal; [email protected] a r X i v : . [ m a t h . A P ] M a y n fact, in this setting the more general notion of linearity is pq = 1 or, equivalently, 1 / ( p + 1) +1 / ( q + 1) = 1 [14]. On the other hand, the last inequality in (1.2) means that the exponents ( p, q )are below the critical hyperbola ( i.e. , ( p, q ) is subcritical) [13, 26], and this condition is triviallysatisfied if N = 1 , pq < etc . We refer to the surveys [6, 15, 31] for an overviewof the topic and to [8, 11, 12, 24] for more recent results. Most of these papers use Dirichletboundary conditions and, up to our knowledge, the few papers addressing Neumann problemsare [4,27,29,37], where existence of positive solutions and concentration phenomena are studied,and [9], which centers on existence of positive radial solutions. However, these papers focus ona different operator of the form Lw = − ∆ w + V ( x ) w , with V positive. In comparison with(1.1), the shape of solutions changes drastically; for instance, the operator L with Neumann b.c.induces a norm, and this allows the existence of positive solutions, while all nontrivial solutionsof (1.1) are sign-changing . Indeed, if ( u, v ) is a classical solution of (1.1), then by the Neumannb.c. and the divergence theorem, (cid:90) Ω | u | p − u = (cid:90) Ω | v | q − v = 0 . (1.3)Since u ≡ v ≡
0, (1.3) is only satisfied if ( u, v ) is trivial or if both components aresign-changing. Condition (1.3) is called a compatibility condition . As far as we know, our paperis the first to study problem (1.1).We remark that if p = q > u, v ) is a classical solution of (1.1), then u ≡ v in Ω and(1.1) reduces to the scalar equation − ∆ u = | u | p − in Ω with ∂ ν u = 0 on ∂ Ω, see Lemma 2.6.Therefore, all our results cover the single equations case.Condition (1.2) together with Sobolev embeddings and the Rellich–Kondrachov theorem im-plies that W , p +1 p (Ω) (cid:44) → L q +1 (Ω) and W , q +1 q (Ω) (cid:44) → L p +1 (Ω) compactly . (1.4)A strong solution of (1.1) is defined as a pair ( u, v ) ∈ W , q +1 q (Ω) × W , p +1 p (Ω) satisfying theequations a.e. in Ω, and the boundary conditions in the trace sense. Problem (1.1) has avariational structure, and (1.1) are the Euler-Lagrange equations of the energy functional( u, v ) (cid:55)→ I ( u, v ) = (cid:90) Ω ∇ u · ∇ v − | u | p +1 p + 1 − | v | q +1 q + 1 dx. (1.5)We define a least energy (nodal) solution as a nontrivial strong solution of (1.1) achieving thelevel c := inf { I ( u, v ) : ( u, v ) (cid:54)≡ (0 , , ( u, v ) is a strong solution of (1.1) } . (1.6)In view of (1.2) and (1.4), the functional I is well defined at strong solutions. Our main resultis concerned with existence, regularity, and qualitative properties of least energy solutions. Theorem 1.1.
Let N ≥ , Ω ⊂ R N be a smooth bounded domain, and let p and q satisfy (1.2) . The set of least energy solutions is nonempty. If ( u, v ) is a least energy solution, then ( u, v ) ∈ C ,ε (Ω) × C ,ε (Ω) is a classical solution of (1.1) and the following holds.(i) (Monotonicity in 1D) If N = 1 and Ω = ( − , , then u (cid:48) v (cid:48) > in Ω ; in particular, u and v are both strictly monotone increasing or both strictly monotone decreasing in Ω . ii) (Symmetry & symmetry breaking) If N ≥ and Ω = B (0) or Ω = B (0) \ B δ (0) for some δ ∈ (0 , , then there is e ∈ ∂B (0) such that u and v are foliated Schwarz symmetric withrespect to e and the functions u and v are not radially symmetric.(iii) (Unique continuation property) If pq < , then the zero sets of u and v have zero Lebesguemeasure, i.e. , |{ x ∈ Ω : u ( x ) = 0 }| = |{ x ∈ Ω : v ( x ) = 0 }| = 0 . Results for the scalar equation follow immediately from Theorem 1.1, see Corollary 1.4 below.Our approach to show Theorem 1.1 is based on a variant of the dual method [3,14]. Later in thisintroduction we motivate the use of this approach and also the relationship between our resultsand previously known results for the single-equation problem. To describe the dual framework,we introduce some notation used throughout the whole paper. Let p and q satisfy (1.2) and, for s >
1, let X s = (cid:110) f ∈ L s (Ω) : (cid:90) Ω f = 0 (cid:111) , α := p + 1 p , β := q + 1 q , and X := X α × X β , endowed with the norm (cid:107) ( f, g ) (cid:107) X = (cid:107) f (cid:107) α + (cid:107) g (cid:107) β . Let K denote the inverse (Neumann) Laplaceoperator with zero average, that is, if h ∈ X s (Ω), then u := Kh ∈ W ,s (Ω) is the unique strongsolution of − ∆ u = h in Ω satisfying ∂ ν u = 0 on ∂ Ω and (cid:82) Ω u = 0, see Lemma 2.1 below. In thissetting, the (dual) energy functional φ : X → R is given by φ ( f, g ) := (cid:90) Ω | f | α α + | g | β β − g Kf dx, ( f, g ) ∈ X. (1.7)Since (1.3) holds for any nontrivial solution, we require a suitable translation of K . For t > K t : X t +1 t → W , t +1 t (Ω) be given by K t h := Kh + c t ( h ) for some c t ( h ) ∈ R such that (cid:90) Ω | K t h | t − K t h = 0 . Then, a critical point ( f, g ) of φ solves the dual system K q f = | g | q − g and K p g = | f | p − f in Ω,see Lemma 2.3.In the sublinear case ( pq < φ achieves its global minimum in X (see Lemma2.8); whereas, in the superlinear case ( pq > φ is unbounded from below, but it can beminimized (see Lemma 2.10) in the Nehari-type set N := { ( f, g ) ∈ X \{ (0 , } : φ (cid:48) ( f, g )( γ f, γ g ) = 0 } with γ := βα + β , γ := αα + β . Therefore, we often refer to a minimizer ( f, g ) ∈ X satisfying φ ( f, g ) = inf X φ if pq < φ ( f, g ) = inf N φ if pq > . (1.8)In particular, if ( f, g ) satisfies (1.8), then ( u, v ) := ( K p g, K q f ) is a least energy solution , that is, φ ( f, g ) = I ( u, v ) = c , with c as in (1.6) and solves (1.1), see Lemmas 2.5, 2.8, and 2.10 below.We now describe in more detail the different techniques involved in the proof of Theorem 1.1.The existence is obtained using the subcriticality assumption (1.2) and the compactness of theoperator K , while the regularity follows from a bootstrap argument (Proposition 2.4). The uniquecontinuation property is shown by extending the techniques from [25, Section 3] to the settingof Hamiltonian elliptic systems and to the dual-method framework (later in this introduction3e compare in more detail the results and strategies from [25] with ours). Moreover, the factthat least energy solutions are foliated Schwarz symmetric is due to a characterization of sets offunctions with this symmetry in terms of invariance under polarizations, see Lemma 4.7 belowand [5, 8, 10, 25, 32] for similar results in other settings.One of our main contributions, from the methodological point of view, is the proof of thestrict monotonicity of solutions (for N = 1) and of the symmetry-breaking phenomenon ( i.e. ,that least energy solutions are not radial). These results require first a deep understanding ofthe radial setting: consider Ω ⊂ R N to be eitherΩ = B (0) and fix δ := 0 (for any N ≥
1) orΩ = B (0) \ B δ (0) for some δ ∈ (0 ,
1) (for any N ≥ . (1.9)If Y is a set of functions, we use Y rad to denote the subset of radial functions in Y and we call( u, v ) := ( K p g, K q f ) a least energy radial solution if ( f, g ) satisfies φ ( f, g ) = inf X rad φ if pq < φ ( f, g ) = inf N rad φ if pq > . (1.10)Our next result shows that these radial solutions are remarkably rigid. We denote the radialderivative of w by w r and use a slight abuse of notation setting w ( | x | ) = w ( x ). Theorem 1.2.
Let p, q satisfy (1.2) and Ω , δ as in (1.9) . The set of least energy radial solutionsis nonempty and, if ( u, v ) is a least energy radial solution, then ( u, v ) ∈ C ,ε (Ω) × C ,ε (Ω) is aclassical radial solution of (1.1) and u r v r > in ( δ, ; in particular, u and v are both strictlymonotone increasing or both strictly monotone decreasing in the radial variable. The existence and regularity follows similarly as in Theorem 1.1. The strict monotonicityrelies on the following new transformation denoted with a superindex (cid:62) : let Ω, δ as in (1.9) and I : L ∞ rad (Ω) → C rad (Ω) , I h ( x ) := (cid:90) { δ< | y | < | x |} h ( y ) dy = N ω N (cid:90) | x | δ h ( ρ ) ρ N − dρ, F : C rad (Ω) → L ∞ rad (Ω) , F h := ( χ {I h> } − χ {I h ≤ } ) h. Then, for h ∈ C rad (Ω), the (cid:62) -transformation of h is given by h (cid:62) ∈ L ∞ rad (Ω) , h (cid:62) ( x ) := ( F h ) ( ω N | x | N − ω N δ N ) . where ω N = | B | is the volume of the unitary ball in R N . The function F h can be seen as a suitable flippingof h on the set {I h ≤ } , and this step is very important to construct monotone decreasingsolutions. In Remark 3.5 we motivate further the definition of h (cid:62) and in Figure 1 in Section 3.2we illustrate the construction of h (cid:62) in a ball and compare it with the Schwarz symmetrization h ∗ ( x ) = h ( ω N | x | N ).Observe that annuli, sign-changing functions, and Neumann boundary data are non-standardconditions to work with rearrangements; in fact, many results relying on symmetrizations failin these settings; for example, the standard Polya-Szeg˝o inequality only holds for nonnegativefunctions with zero boundary data. Our approach is able to cover these cases mainly becauseof two reasons: the radiality assumption and the fact that we use Lebesgue spaces within adual framework, which gives us more flexibility in the construction of our transformation; inthis sense, we are transforming the dual variables ( f, g ) to obtain, together with variationaltechniques, monotonicity information of solutions ( u, v ) := ( K p g, K q f ).In general, h (cid:62) is not a (level-set) rearrangement of h , since the maximum value of h may varydue to the flipping F h ; however, L t -norms and (zero) averages are preserved (see Proposition3.4) and the transformed functions have less energy, as stated in the following result.4 heorem 1.3. Let p, q > , Ω as in (1.9) , and let f, g : Ω → R be continuous and radiallysymmetric functions with (cid:82) Ω f = (cid:82) Ω g = 0 . Then ( f (cid:62) , g (cid:62) ) ∈ X and φ ( f (cid:62) , g (cid:62) ) ≤ φ ( f, g ) . (1.11) Furthermore, if f and g are nontrivial and φ ( f (cid:62) , g (cid:62) ) = φ ( f, g ) , then f and g are monotone inthe radial variable and if ( u, v ) := ( K p g, K q f ) , then u and v are radially symmetric and strictlymonotone in the radial variable. The proof exploits the one-dimensionality of the problem and uses elementary rearrangementtechniques. Note that, if Ω is a ball, then h (cid:62) is actually the Schwarz symmetrization of F h . Theflipping F , however, is necessary, since it can be shown that (1.11) does not hold in general usingmerely the Schwarz symmetrization (even in the scalar case p = q ), see Remark 3.6 below.Theorem 1.3 is the main tool to show the monotonicity claims in Theorem 1.1 and Theorem1.2. These results are of independent interest and are new even in the single-equation case (1.15).Furthermore, Theorem 1.3 is also the starting point for the proof of the symmetry-breakingphenomenon, which, in the dual setting, reads as follows.If ( f, g ) satisfies (1.8), then f and g are not radial . (1.12)The proof of this claim is done by contradiction: if ( f, g ) is radially symmetric and ( u, v ) :=( K p g, K q f ), then ( u x , v x ) is a strong (Dirichlet) solution of − ∆ u x = q | v | q − v x =: ¯ g, − ∆ v x = p | u | p − u x =: ¯ f in Ω with u x = v x = 0 on ∂ Ω . Therefore, from the minimality of ( f, g ) (approximating φ (cid:48)(cid:48) in a suitable sense) we infer that (cid:90) Ω( e ) p | u | p − u x ( u x − K ¯ g ) + q | v | q − v x ( v x − K ¯ f ) dx ≥ , Ω( e ) := Ω ∩ { x > } . (1.13)Here, Theorem 1.2 is very important to control the (possibly singular) terms | u | p − and | v | p − ,since the strict monotonicity implies that the nodal sets of u and v are merely two inner spheres.In the end, a contradiction is obtained by showing—using the radiality assumption, maximumprinciples, and Hopf’s boundary point Lemma—that the Neumann solution ( K ¯ g, K ¯ f ) dominatesthe Dirichlet solution ( u x , v x ) in Ω( e ), and this would imply that the integral in (1.13) is strictlynegative. Observe that the symmetry-breaking statement in Theorem 1.1 follows directly from(1.12). For other symmetry-breaking results for single equations we refer to [1] for Dirichletboundary conditions, to [19,25] for Neumann boundary conditions, and to [6] for a (perturbative)symmetry-breaking result for Dirichlet Hamiltonian systems. See also the survey [35] and thereferences therein.We now focus on the particular case p = q , where (1.2) reduces to p > , p (cid:54) = 1 , ( N − p < ( N + 2) . (1.14)In this situation, we show in Lemma 2.6 below that any classical solution ( u, v ) of (1.1) satisfies u ≡ v , and problem (1.1) is equivalent to − ∆ u = | u | p − u in Ω , ∂ ν u = 0 on ∂ Ω , (1.15)whose solutions are critical points of E ( u ) := I ( u, u )2 = (cid:90) Ω |∇ u | − p | u | p dx (1.16)Then, as a particular case of Theorems 1.1 and 1.2 we have the following.5 orollary 1.4. Let N ≥ , Ω ⊂ R N be a smooth bounded domain, and let p satisfy (1.14) . Theset of least energy solutions of (1.15) is nonempty and it is contained in C ,ε (Ω) .(i) (Unique continuation) If p < , then the zero set of every least energy solution has zeroLebesgue measure.(ii) (Monotonicity, symmetry, and symmetry breaking) If N = 1 and Ω = ( − , , then everyleast energy solution is strictly monotone in Ω . If N ≥ and Ω is either a ball or anannulus as in (1.9) , then every least energy solution is foliated Schwarz symmetric and itis not radially symmetric.(iii) (Radial solutions) Let Ω be a ball or an annulus as in (1.9) , then the set of least energy radial solutions is nonempty. If u is a least energy radial solution, then u ∈ C ,ε (Ω) is aclassical radial solution of (1.14) and u is strictly monotone in the radial variable. Up to our knowledge, the monotonicity of least energy radial solutions (part ( iii ) of Corollary1.4) is new. Part (ii) is also new in the subcritical superlinear regime as well as the symmetrybreaking result for annuli in the sublinear case. In this sense, this corollary complements theresults in [25], where unique continuation, symmetry, and symmetry breaking for least energysolutions of (1.15) in the case 0 < p < p = 0 is also considered in [25],interpreting (1.15) as − ∆ u = sign( u )). Note that the unique continuation property when p > < p < N p = { u ∈ H (Ω) : (cid:90) Ω | u | p − u = 0 } , (1.17)proving that the least energy level is achieved (note that N p is not a C -manifold because p < N ≥
3, the (direct) functional I , given in (1.5), is not well defined in H := H (Ω) × H (Ω) under (1.2). In fact, even working on a range of ( p, q ) where I is welldefined in H (or using (1.2) with p, q > I is strongly indefinite , in the sense that the principal part (cid:82) Ω ∇ u · ∇ v does not have a sign,and it is actually positive in an infinite dimensional subspace, and negative in another. Tocontrol this difficulty, a direct approach is based, for example, on abstract linking theoremsor on special Nehari-type sets. Different alternatives are available in the literature to studyHamiltonian systems variationally (see, e.g. , the survey [6]), each one with its own advantagesand disadvantages. For example, a common strategy is to reduce the system to a single higher-order problem; however, it is not clear how to study Neumann b.c. in this setting and usinghigher-order Sobolev spaces brings additional complications (for the use of rearrangements, forexample).Dual methods, on the other hand, offer a flexible and elegant alternative. This approachentails the challenge of controlling the effects of the nonlocal operator K in the functional φ ; butit compensates this difficulty with many advantages, for example, the compatibility conditions(1.3) translate to (cid:82) Ω f = (cid:82) Ω g = 0 in the dual formulation; in particular, this allows in thesublinear case to minimize and differentiate the functional φ in the Banach space X (recall thatin [25] the functional (1.16) is minimized on (1.17), which is not a manifold); whereas, in thesuperlinear case, the dual formulation allows to minimize in a Nehari manifold in a Banach space.6he use of a direct or a dual approach has, of course, a strong influence on the methods tostudy qualitative properties of solutions; this is particularly clear in the proof of the symmetrybreaking, described above, where our proof relies on a transformation in dual spaces to obtainmonotonicity and then on comparison principles to control the effects of the nonlocal operator K . The proof of the symmetry breaking result in [25] (although is also done by contradictionfinding a direction along which the energy would decrease) is very different from ours, and itrelies on a unique continuation property for minimizers, which is obtained using known uniquenessresults [36] and nonoscillation criteria [20] for sublinear ODEs of type y (cid:48)(cid:48) + a ( t ) | y | p − y = 0. Thesegeneral theorems are not known for systems of ODEs, and in fact, they may fail in general. Usingelementary manipulations in an ODE setting, one can find some extensions of these techniquesto systems. Although we do not need these results for any of our proofs, we believe they can beof independent interest; in particular, we use them to show a result on the simplicity of zeros of any radial solution that satisfies a unique continuation property. Theorem 1.5.
Let p, q > , N ≥ , Ω , δ as in (1.9) , and ( u, v ) ∈ [ C ,ε (Ω)] be a radial classicalsolution of (1.1) such that u − (0) ∩ v − (0) has empty interior. Then u, v ∈ W := { w ∈ C (Ω) : ∇ w ( x ) (cid:54) = 0 if x ∈ Ω satisfies | x | > δ and w ( x ) = 0 } , (1.18)This theorem yields that, if ( u, v ) is a radial solution of (1.1) satisfying a (weak) uniquecontinuation property, then the nodal set of u and of v is at most a countable union of spheresand thus | u − (0) | = | v − (0) | = 0.To close this introduction, we mention some open questions about system (1.1): it is unknownwhether or not least energy solutions are (up to rotations) unique ; in fact, uniqueness is open evenfor least energy radial solutions. Moreover, when pq < i.e. , that the nodal sets of solutions have zero Lebesgue measure) holds in general for all solutions, and not just for minimizers. When pq >
1, with the exception of the single equationcase p = q , the unique continuation property is an open question even for least energy solutions.In Remark 5.5 below we observe that, in the superlinear case, if a least energy solution ( u, v )vanishes on a set of positive measure, then it has a zero of infinite order. For single equations oftype − ∆ u + V ( x ) u = 0 such a result is available for a large class of potentials (see [16, Proposition3] and [21, Proposition 1.1]) and having a zero of infinite order typically implies that a solutionis identically zero (see [22]).The paper is organized as follows. In Section 2 we provide some general auxiliary results inour dual framework; in particular, we show Lemmas 2.8 and 2.10, which prove the existence andregularity claims in Theorem 1.1, as well as Lemma 2.6 which shows (together with Theorems 1.1and 1.2) Corollary 1.4. Section 3 is devoted to the study of monotonicity properties of radialminimizers, here the full details of the construction of the (cid:62) -transformation can be found aswell as the proof of Theorems 1.2 and 1.3. These results are then used in Section 4 to provethe symmetry breaking result in Theorem 1.1 as well as the foliated Schwarz symmetry claim.Finally, in Section 5 we show Theorem 1.5 as well as the unique continuation statement inTheorem 1.1. 7 .1 Notation Throughout the paper p and q always denote the exponents in (1.1). We divide our proofs intwo groups: Sublinear: pq < , (1.19)Superlinear and subcritical: pq > p + 1 + 1 q + 1 > N − N . (1.20)Note that pq < pq < / ( p + 1) +1 / ( q + 1) > > ( N − /N . If N = 1 , p, q > α := p + 1 p , α (cid:48) = p + 1 , β := q + 1 q , β (cid:48) = q + 1 . (1.21)Note that α (cid:48) and β (cid:48) are the corresponding conjugate exponents of α and β , that is α − +( α (cid:48) ) − =1 and β − + ( β (cid:48) ) − = 1.Moreover, we define X , X s , K , and K s as in the Introduction. For s ≥ (cid:107) · (cid:107) s and (cid:107) · (cid:107) ,s arethe standard norms in L s (Ω) and W ,s (Ω) respectively and, if s > N > s , thenwe denote by s ∗ := NsN − s > s the critical Sobolev exponent, that is, s ∗ is the biggest exponentwhich allows the (continuous) embedding W ,s (Ω) (cid:44) → L s ∗ (Ω). If N = 1 , s ∗ = ∞ .If Y is a vector space, then [ Y ] := Y × Y . We denote by B r the ball in R N of radius r > u : Ω → R is denoted by u − (0) := { x ∈ Ω : u ( x ) = 0 } .For a measurable set A ⊂ R N , | A | denotes its Lebesgue measure, and the function χ A is thecharacteristic function of A , that is, χ A ( x ) = 1 if x ∈ A and χ A ( x ) = 0 if x (cid:54)∈ A . Finally, we use ω N to denote the measure of the unitary ball in R N . In this section we describe our dual approach. In the following Ω is a smooth bounded domainin R N , p and q satisfy (1.2), and let X s , X , K , K t , α , and β be as above. The following lemma recalls some well-known regularity for Neumann problems, see for example[30, Theorem and Lemma in page 143] (see also [2, Theorem 15.2]).
Lemma 2.1. If s > , Ω be a smooth bounded domain in R N , and h ∈ L s (Ω) with (cid:82) Ω h = 0 ,then there is a unique strong solution u ∈ W ,s (Ω) of − ∆ u = h in Ω , ∂ ν u = 0 on ∂ Ω , (cid:90) Ω u = 0 (2.1) in particular, (cid:90) Ω ∇ u ∇ ϕ = (cid:90) Ω hϕ for all ϕ ∈ W ,s (cid:48) (Ω) , s (cid:48) = ss − , and there is C (Ω , s ) = C > such that (cid:107) u (cid:107) ,s ≤ C (cid:107) h (cid:107) s . f ∈ L s (Ω) for some s >
1, then Kf is the solution of (2.1) with h = f . Observethat (cid:90) Ω g K q f = (cid:90) Ω g Kf = (cid:90) Ω Kg f = (cid:90) Ω K p g f, (2.2)by integration by parts and because (cid:82) Ω f = (cid:82) Ω g = 0. Our next result shows that φ is continuouslydifferentiable. Lemma 2.2.
The functional φ : X → R as defined in (1.7) is continuously differentiable in X with φ (cid:48) ( f, g )( ϕ, ψ ) = (cid:90) Ω | f | α − f ϕ + | g | β − gψ − ψ Kf − ϕ Kg dx for ( f, g ) , ( ϕ, ψ ) ∈ X. (2.3) Proof.
Let φ = ψ − T , whereΨ( f, g ) := (cid:90) Ω | f | α α + | g | β β dx and T ( f, g ) := (cid:90) Ω g Kf, ( f, g ) ∈ X. (2.4)Since α, β > (cid:48) ( f, g )( ϕ, ψ ) = (cid:90) Ω | f | α − f ϕ + | g | β − gψ dx, ( f, g ) , ( ϕ, ψ ) ∈ X. (2.5)Now we show that T is bilinear and bounded. Indeed, this follows directly from the fact that K ( λh ) = λKh for h ∈ L s (Ω), s ∈ { α, β } , and the following integration by parts T ( f, g ) = (cid:90) Ω gKf = (cid:90) Ω f Kg, ( f, g ) ∈ X. (2.6)Moreover, by H¨older’s inequality, Lemma 2.1, and the first embedding in (1.4), there are C , C > | T ( f, g ) | ≤ (cid:107) g (cid:107) β (cid:107) Kf (cid:107) β (cid:48) ≤ C (cid:107) g (cid:107) β (cid:107) Kf (cid:107) W ,α (Ω) ≤ C (cid:107) g (cid:107) β (cid:107) f (cid:107) α . (2.7)Thus T is a (continuously differentiable) bilinear bounded mapping and, by (2.6), T (cid:48) ( f, g )( ϕ, ψ ) = (cid:90) Ω ψ Kf + ϕ Kg dx, ( f, g ) , ( ϕ, ψ ) ∈ X. (2.8)Therefore, (2.3) follows from (2.5) and (2.8).Before we argue existence of solutions, we establish a one-to-one relationship between criticalpoints of φ and strong solutions of (1.1), see [6, Proposition 3.1] for the Dirichlet case. Lemma 2.3.
An element ( f, g ) ∈ X is a critical point of φ , i.e., φ (cid:48) ( f, g )( ϕ, ψ ) = 0 for all ( ϕ, ψ ) ∈ X, (2.9) if and only if ( u, v ) := ( K p g, K q f ) ∈ W ,β (Ω) × W ,α (Ω) is a strong solution of (1.1) , that is, ( u, v ) solves (1.1) a.e. in Ω . roof. Let ( f, g ) ∈ X satisfy (2.9) and let ( u, v ) := ( K p g, K q f ). Then, using (2.2), (cid:90) Ω ( | f | α − f − u ) ϕ + ( | g | β − g − v ) ψ dx = 0 for all ( ϕ, ψ ) ∈ X = X α × X β . Let w := | f | α − f − u and, for a function ζ ∈ L α (Ω), set ζ := | Ω | − (cid:82) Ω ζ ∈ R . Then ( ζ − ζ ) ∈ X α and0 = (cid:90) Ω w ( ζ − ζ ) = (cid:90) Ω wζ − ζ (cid:90) Ω w = (cid:90) Ω wζ − w (cid:90) Ω ζ = (cid:90) Ω ( w − w ) ζ for all ζ ∈ L α (Ω) , and therefore w − w = 0 a.e. in Ω, which implies that | f | α − f = u + w a.e. in Ω and thus f = | u + w | p − ( u + w ) a.e. in Ω. Furthermore, since ( f, g ) ∈ X and u = K p g , we havethat (cid:82) Ω | u | p − u = 0 = (cid:82) Ω f = (cid:82) Ω | u + w | p − ( u + w ), which implies that w = 0, and therefore − ∆ v = − ∆( K q f ) = f = | u | p − u a.e. in Ω. Analogously, − ∆ u = | v | q − v a.e. in Ω, as claimed.For the converse implication, let ( u, v ) := ( K p g, K q f ) ∈ W ,β (Ω) × W ,α (Ω) be a strongsolution of (1.1) for some ( f, g ) ∈ X . Then, necessarily f = | u | p − u and g = | v | q − v a.e. in Ω,which implies that u = | f | α − f and v = | g | β − g a.e. in Ω, which implies (2.9), by (2.2).The next Proposition states that critical points of φ are in fact classical solutions. We referto [33, Theorem 1] and [6, Lemma 5.16] for analogous results in the superlinear Dirichlet case. Proposition 2.4.
Let ( f, g ) ∈ X be a critical point of φ , then ( u, v ) := ( K p g, K q f ) ∈ [ C ,ε (Ω)] for some ε > and satisfies (1.1) pointwise.Proof. The proof follows closely [6, Lemma 5.16]. For t, s ≥ W ( s, t ) := W ,s (Ω) × W ,t (Ω) and L ( s, t ) := L s (Ω) × L t (Ω) . Let ( f, g ) ∈ X ⊂ L ( α, β ) be a critical point of φ then, by Lemma 2.3, ( u, v ) ∈ W ( β, α ) (cid:44) → L ( β ∗ , α ∗ ) is a strong solution of (1.1). For n ∈ N , let( β , α ) := ( β, α ) , ( β n +1 , α n +1 ) := ( α ∗ n q , β ∗ n p ) if N > α n , N > β n . Here we are using the notation given in Section 1.1 for the critical Sobolev exponent. Notethat ( | v | q , | u | p ) ∈ L ( α ∗ /q, β ∗ /p ) = L ( β , α ) and then Lemma 2.1 and (1.1) imply ( u, v ) ∈ W ( β , α ) (cid:44) → L ( β ∗ , α ∗ ). But then ( | v | q , | u | p ) ∈ L ( β , α ), which gives ( u, v ) ∈ W ( β , α ) (cid:44) → L ( β ∗ , α ∗ ). Iterating this procedure we obtain that ( u, v ) ∈ W ( β n , α n ) as long as N > α n − and N > β n − .We claim that N ≤ s for some s ∈ { α n , β n : n ∈ N } , (2.10)and then the proposition follows from Lemma 2.3, the embedding W ,s (Ω) (cid:44) → C µ (Ω) for some µ >
0, and standard Schauder estimates (see [18, Theorem 6.31] and the Remark after thetheorem), since t (cid:55)→ | t | r − t for r > N > s for all s ∈ { α n , β n : n ∈ N } and considerthe sequence S n := ( qβ n , pα n ). By the subcriticality assumption we have that W ( β, α ) (cid:44) → L ( α (cid:48) , β (cid:48) ) = L ( pα, qβ ) and therefore S = ( qβ , pα ) = ( α ∗ , β ∗ ) > ( qβ, pα ) = S and ( S n ) n ∈ N l , l > S n → ( l , l ) and note that, by the monotonicity, l > pα = p + 1 . (2.11)Moreover( l , l ) = lim n →∞ ( α ∗ n , β ∗ n ) = ( N l N p − l , N l N q − l ) , which implies l = N ( pq − q + 1) . Since the function t (cid:55)→ h ( t ) := N ( pt − t +1) is increasing for t > q < t := p + N +2 pN − p − , by thesubcriticality condition (1.20), we have that l = h ( q ) < h ( t ) = p + 1 , which contradicts (2.11).Then (2.10) holds and this ends the proof.The next Lemma shows the relationship between the dual and the direct energy functionalswhen evaluated on solutions . Lemma 2.5.
Let ( f, g ) ∈ X be a critical point of φ and let ( u, v ) := ( K p g, K q f ) . Then φ ( f, g ) = I ( u, v ) := (cid:90) Ω ∇ u ∇ v − | u | p +1 p + 1 − | v | q +1 q + 1 dx Proof.
Let f, g, u and v as in the statement. By Proposition 2.4, we have that ( u, v ) is a solutionof (1.1) and, integrating by parts, φ ( f, g ) = (cid:90) Ω | f | α α + | g | β β − gKf dx = (cid:90) Ω p | u | p +1 p + 1 + q | v | q +1 q + 1 − ∇ u ∇ v dx. Therefore φ ( f, g ) − I ( u, v ) = (cid:82) Ω | u | p +1 + | v | q +1 − ∇ u ∇ v dx = 0, by (1.1).We finish this Section with a result in the case p = q , in which (1.1) reduces to a singleequation. The proof is the same as in the superlinear Dirichlet case [8, Theorem 1.5] and weinclude it for the reader’s convenience. Lemma 2.6.
Let p = q > and ( u, v ) ∈ C (Ω) × C (Ω) be a classical solution to (1.1) , then u = v .Proof. By testing (1.1) with u, v and integrating by parts we have that (cid:90) Ω |∇ u | = (cid:90) Ω | v | p − v u, (cid:90) Ω |∇ v | = (cid:90) Ω | u | p − u v, and (cid:107) v (cid:107) α (cid:48) α (cid:48) = (cid:90) Ω ∇ u ∇ v = (cid:107) u (cid:107) α (cid:48) α (cid:48) . Then, by H¨older’s inequality, (cid:90) Ω |∇ u | + (cid:90) Ω |∇ v | ≤ (cid:107) u (cid:107) α (cid:48) (cid:107) v (cid:107) pα (cid:48) + (cid:107) v (cid:107) α (cid:48) (cid:107) u (cid:107) pα (cid:48) = 2 (cid:90) Ω ∇ u ∇ v, which implies ∇ u = ∇ v in Ω and then v − u ≡ c in Ω for some constant c ∈ R . Therefore, by(1.1), | u | p − u = − ∆ u = − ∆( u + c ) = | u + c | p − ( u + c ) in B . If u ≡ v ≡
0. If u (cid:54)≡ u changes sign and there is x ∈ B with u ( x ) = 0, but then | c | p − c = 0, and u = v in Ω asclaimed. 11 .2 Existence of least energy solutions: sublinear case Assume (1.19), that is, p, q > pq <
Lemma 2.7.
The functional φ : X → R as defined in (1.7) is coercive.Proof. Let ε := min { α − , β − } . Then from (2.7), Young’s inequality, and the fact that β (cid:48) < α (since pq <
1) there is C ( α, β ) = C > | T ( f, g ) | ≤ ε ( (cid:107) f (cid:107) αα + (cid:107) g (cid:107) ββ ) + C . This yieldsthat φ ( f, g ) ≥ ε ( (cid:107) f (cid:107) αα + (cid:107) g (cid:107) ββ ) − C , (2.12)that is, φ is coercive. Lemma 2.8.
The functional φ achieves a negative minimum in X , that is, min X φ = φ ( f, g ) < for some ( f, g ) ∈ X. Moreover, φ (cid:48) ( f, g ) = 0 in X , ( u, v ) := ( K p g, K q f ) ∈ [ C ,ε (Ω)] is a classical solution of (1.1) ,and ( u, v ) is a least energy solution, that is, I ( u, v ) = c with c as in (1.6) .Proof. For n ∈ N let x n := ( f n , g n ) be a minimizing sequence in X , which exists by Lemma 2.7.Since X is reflexive there is x := ( f, g ) ∈ X such that x n (cid:42) x weakly in X and ( x n ) is bounded in X in virtue of (2.12). Since the operator K is compact by Lemma 2.1 and the compact embedding(1.4), we have that K ( f n ) converges strongly to Kf in L β (cid:48) (Ω). Using the lower semicontinuityof norms we have that min X φ = lim inf n →∞ φ ( f n , g n ) ≥ φ ( f, g ), and therefore φ achieves its minimum.To see that the minimum is strictly negative, let T and Ψ as in (2.4) and let ϕ ∈ C ∞ c (Ω) \{ } such that (cid:82) Ω ϕ = 0. Then ( ϕ, ϕ ) ∈ X \{ (0 , } and T ( ϕ, ϕ ) = (cid:90) Ω ϕKϕ = (cid:90) Ω ( − ∆ u ) u = (cid:90) Ω |∇ u | > , where u := Kϕ.
Thus, since pq < αβ > α + β , φ ( t βα + β ϕ, t αα + β ϕ ) = t αβα + β Ψ( ϕ, ϕ ) − tT ( ϕ, ϕ ) < t > φ (cid:48) ( f, g ) = 0 in X by Lemma 2.2, ( u, v ) := ( K p g, K q f ) ∈ [ C ,ε (Ω)] is a classical solutionof (1.1), by Proposition 2.4, and ( u, v ) is a least energy solution, by Lemma 2.5. Let p, q ∈ (0 , ∞ ) such that (1.20) holds. γ := βα + β , γ := αα + β , and γ := αβα + β < , (2.13)where γ < pq >
1. In particular, γ α = γ β = γ and γ + γ = 1. We define theNehari-type set N := { ( f, g ) ∈ X \{ (0 , } : φ (cid:48) ( f, g )( γ f, γ g ) = 0 } = (cid:26) ( f, g ) ∈ X \{ (0 , } : (cid:90) Ω γ | f | α + γ | g | β dx = (cid:90) Ω f Kg (cid:27) . (2.14)12 emma 2.9. Let ( f, g ) ∈ X \{ (0 , } such that (cid:82) Ω f Kg > , then t := t ( f, g ) := (cid:16) γ (cid:82) Ω | f | α + γ (cid:82) Ω | g | β (cid:82) Ω f Kg (cid:17) − γ (2.15) is the unique maximum of the function s (cid:55)→ φ ( s γ f, s γ g ) and ( t γ f, t γ g ) ∈ N .Proof. Since φ ( s γ f, s γ ) = s γ (cid:82) Ω | f | α α + | g | β β dx − s (cid:82) Ω f Kg, γ <
1, and (cid:82) Ω f Kg >
0, the map s (cid:55)→ φ ( s γ f, s γ g ) has a unique positive critical point, which is a global maximum. The claimnow follows by direct computations. Lemma 2.10.
There is ( f, g ) ∈ N such that φ ( f, g ) = inf N φ . Moreover, φ (cid:48) ( f, g ) = 0 in X , ( u, v ) := ( K p g, K q f ) ∈ [ C ,ε (Ω)] is a classical solution of (1.1) , and ( u, v ) is a least energysolution, that is, I ( u, v ) = c with c as in (1.6) .Proof.
1) The functional φ is bounded from below on N , since φ ( f, g ) = 1 − γα (cid:90) Ω | f | γ + 1 − γβ (cid:90) Ω | g | β ≥ f, g ) ∈ N . (2.16)Thus inf N φ ∈ R . Take a minimizing sequence ( f n , g n ) ∈ N . From (2.16), we have that (cid:107) f n (cid:107) α and (cid:107) g n (cid:107) β are bounded and thus, up to a subsequence, f n (cid:42) f in L α , g n (cid:42) g in L β , for some( f, g ) ∈ X .2) Recalling (2.7) and using Young’s inequality, we have γ (cid:107) f n (cid:107) αα + γ (cid:107) g n (cid:107) ββ = (cid:90) Ω f n Kg n ≤ C (cid:107) f n (cid:107) α (cid:107) g n (cid:107) β ≤ C ( γ (cid:107) f n (cid:107) γ α + γ (cid:107) g n (cid:107) γ β )for some C > γ > α , γ > β , there exists δ > (cid:82) Ω f n Kg n ≥ δ for every n .3) Combining this with the properties of weak convergence and the compactness of the operator K , we have ( f, g ) (cid:54) = (0 ,
0) and, by definition of N ,0 < (cid:90) Ω γ | f | α + γ | g | β dx ≤ (cid:90) Ω f Kg. Therefore, by Lemma 2.9, there exists 0 < t ≤ t γ f, t γ g ) ∈ N . Then,inf N φ ≤ φ ( t γ f, t γ g ) = t γ (1 − γ ) (cid:90) Ω | f | α α + | g | β β dx ≤ (1 − γ ) (cid:90) Ω | f | α α + | g | β β dx ≤ lim inf n →∞ φ ( f n , g n ) = inf N φ, thus t = 1, and ( f, g ) ∈ N and achieves inf N φ .4) Defining τ ( h, k ) = φ (cid:48) ( h, k )( γ h, γ k ) for ( h, k ) ∈ N , we have τ (cid:48) ( h, k )( γ h, γ k ) = γ ( γ − (cid:90) Ω | h | α + γ ( γ − (cid:90) Ω | k | β < . Therefore N is a manifold and, since ( f, g ) achieves inf N φ , then by Lagrange’s multiplier rulethere exists λ ∈ R such that φ (cid:48) ( f, g ) = λτ (cid:48) ( f, g ). By testing this identity with ( γ f, γ g ), we seethat actually λ = 0 and ( f, g ) is a critical point of φ in X . Thus ( u, v ) := ( K p g, K q f ) ∈ [ C ,ε (Ω)] is a classical solution of (1.1) by applying Proposition 2.4. Finally, ( u, v ) is a least energy solution,by Lemma 2.5. 13 Monotonicity of radial minimizers
In this subsection Ω ⊂ R N is eitherΩ = B (0) and fix δ := 0 (for any N ≥
1) orΩ = B (0) \ B δ (0) for some δ ∈ (0 ,
1) (for any N ≥ . (3.1)We emphasize that the case B (0) \{ } is not considered in any of our results. Our main goal isto show Theorem 1.2. This result is of independent interest, but it is also instrumental in theproof of the symmetry-breaking result in Theorem 1.1. The proof of Theorem 1.2 is based on anew transformation using rearrangement techniques and a suitable flipping compatible with ourdual method approach. We introduce first some notation and show some preliminary results. Let U ⊂ R N be an open set and h : U → R a measurable function. The decreasing rearrangementof h is given by h : [0 , | U | ] → R , h (0) := ess sup U h h ( s ) := inf { t ∈ R : |{ h > t }| < s } , s > . In particular, h is a non-increasing and left-continuous function [23, Proposition 1.1.1], h is a (level-set) rearrangement of h , which yields in particular that L p -norms and averages arepreserved, that is, (cid:107) h (cid:107) L p ( U ) = (cid:107) h (cid:107) L p (0 , | U | ) and (cid:90) U h = (cid:90) | U | h , (3.2)see [23, Corollary 1.1.3]. By [23, Proposition 1.2.2], we know that if E ⊂ U , then (cid:90) E h ( x ) ≤ (cid:90) | E | h . (3.3)We now focus on the case where U = I := [0 , l ] ⊂ R , for some l >
0. If h : I → R is non-increasingin I , then h = h a.e. in I , see [23, Lemma 1.1.1]. We use the following decomposition for theintegral of a product of two functions. Lemma 3.1 (Particular case of Lemma 1.2.2 in [23]) . Let ϕ, ψ : I → R be measurable functionswith ϕ ∈ L ( I ) , ψ ∈ L ∞ ( I ) , and a, b ∈ R with a ≤ ψ ≤ b . Then (cid:90) I ϕ ψ = a (cid:90) I ϕ + (cid:90) ba (cid:90) { ψ>t } ϕ ( s ) dsdt. A well-known consequence of Lemma 3.1 and (3.3) is the Hardy-Littlewood inequality.
Theorem 3.2 (Particular case of Theorem 1.2.2 in [23]) . Let ϕ ∈ L ( I ) and ψ ∈ L ∞ ( I ) be(possibly sign-changing) functions, then (cid:90) I ϕ ψ ≤ (cid:90) I ϕ ψ (3.4)Under some additional assumptions, the equality case in (3.4) can be used to deduce mono-tonicity properties of the involved functions. 14 emma 3.3. Let ϕ, ψ : I → R with ϕ ∈ L ( I ) and ψ ∈ C ( I ) be a strictly decreasing function.If (cid:82) I ϕ ψ = (cid:82) I ϕ ψ , then ϕ = ϕ a.e. in I .Proof. Let b := ψ (0) and a := ψ ( l ). Then, by Lemma 3.1, a (cid:90) I ϕ + (cid:90) ba (cid:90) { ψ>t } ϕ ( s ) ds dt = (cid:90) I ϕ ψ = (cid:90) I ϕ ψ = a (cid:90) I ϕ + (cid:90) ba (cid:90) { ψ>t } ϕ ( s ) ds dt. (3.5)Since ψ is strictly decreasing and continuous, { ψ > t } = [0 , ψ − ( t )) for every t ∈ [ a, b ]. Therefore,by (3.3), (cid:82) { ψ>t } ϕ ( s ) ds ≤ (cid:82) { ψ>t } ϕ ( s ) ds for all t ∈ ( a, b ) and thus, by (3.5), (cid:90) { ψ>t } ϕ ( s ) ds = (cid:90) { ψ>t } ϕ ( s ) ds for all t ∈ ( a, b ) . (3.6)Furthermore, for every t ∈ I we have { ψ > ψ ( t ) } = [ 0 , t ), because ψ is strictly monotonedecreasing and continuous. Then, (3.6) yields that (cid:82) t ϕ = (cid:82) t ϕ for all t ∈ I , and ϕ = ϕ a.e.in I , by Lebesgue differentiation theorem. In the following we do a slight abuse of notation and use w ( | x | ) = w ( x ) for a radial function w .We use L ∞ rad (Ω) and C rad (Ω) to denote the subspace of radial functions in L ∞ (Ω) and C (Ω),respectively. Let Ω, δ as in (3.1) and let I : L ∞ rad (Ω) → C rad (Ω) , I h ( x ) := (cid:90) { δ ≤| y |≤| x |} h ( y ) dy = N ω N (cid:90) | x | δ h ( ρ ) ρ N − dρ F : C rad (Ω) → L ∞ rad (Ω) , F h := ( χ {I h> } − χ {I h ≤ } ) h. Then, for h ∈ C rad (Ω), the (cid:62) -transformation is given by h (cid:62) ∈ L ∞ rad (Ω) , h (cid:62) ( x ) := ( F h ) ( ω N | x | N − ω N δ N ) . where ω N = | B | is the volumeof the unitary ball in R N . The function F h can be seen as a suitable flipping of h on the set {I h ≤ } , and this step is very important to construct monotone decreasing solutions, whilepreserving the (zero) average and the L p -norm of h , see Proposition 3.4 and Remark 3.5. Seealso Figure 1 below for some examples.In general, the function h (cid:62) is not a (level-set) rearrangement of h , since the maximum valueof h may vary due to F , but it has the following important properties. Proposition 3.4.
Let h : Ω → R be a continuous radial function such that (cid:82) Ω h = 0 . Then, forany t ≥ , (cid:90) Ω h (cid:62) = 0 , (cid:90) Ω | h (cid:62) | t = (cid:90) Ω | h | t , and I ( F h ) = |I h | ≥ in Ω . (3.7) Proof.
The second claim in (3.7) follows by (3.2), the definition of f (cid:62) , the fact that | F h | = | h | in Ω, and polar coordinates, since (cid:90) Ω | h (cid:62) | t = N ω N (cid:90) δ | ( F h ) ( ω N ρ N − ω N δ N ) | t ρ N − dρ = (cid:90) I | ( F h ) | t = (cid:90) Ω | F h | t = (cid:90) Ω | h | t . .2 0.4 0.6 0.8 1.0 - - h ( r ) r - - - I h ( r ) r - F h ( r ) r I ( F h )( r ) r - h (cid:62) ( r ) r - - h ∗ ( r ) r Figure 1: Examples of the functions I h , F h , I ( F h ) = |I h | for a particular radial function h ∈ C ( B (0)), and a comparison between h (cid:62) and the Schwarz symmetrization of h , h ∗ ( r ) := h ( ω N r N ). 16or the last property in (3.7), we claim that, I ( χ {I h> } h ) = χ {I h> } I h and I ( χ {I h ≤ } h ) = χ {I h ≤ } I h in [ δ, . (3.8)Indeed, by continuity and since I h ( δ ) = I h (1) = 0, the set {I h > } is open in ( δ,
1) andtherefore {I h > } = ∪ ∞ i =1 I i , where ( I i ) i ∈ N are disjoint open intervals in ( δ, I i = ( a, b )for some δ ≤ a < b ≤
1, then
N ω N (cid:82) I i h ( s ) s N − ds = I h ( b ) − I h ( a ) = 0, since a, b ∈ ∂ {I h > } .Let r > δ arbitrary and let ζ := max { x : x ∈ [ δ, r ] ∩ {I h ≤ } } ∈ {I h ≤ } (recall that I h ( δ ) = 0 and hence ζ is well defined). If ζ = r then χ {I h> } ( r ) = 0 and {I h > } ∩ [ δ, r ] is open in ( δ, r ), therefore I ( hχ {I h> } )( r ) = N ω N (cid:90) rδ h ( s ) χ {I h> } ( s ) s N − ds = N ω N ∞ (cid:88) i =1 (cid:90) I i ∩ ( δ,r ) h ( s ) s N − ds = 0 = I h ( r ) χ {I h> } ( r ) . If ζ < r , then χ {I h> } ( r ) = 1, I h ( ζ ) = 0, and I ( hχ {I h> } )( r ) = N ω N (cid:90) rδ h ( s ) χ {I h> } ( s ) s N − ds = N ω N (cid:90) rζ h ( s ) s N − ds = I h ( r ) − I h ( ζ ) = I h ( r ) χ {I h> } ( r ) . Thus the first equality in (3.8) follows, but then, by the linearity of the integral, I ( h χ {I h ≤ } ) = I ( h ) − I ( h χ {I h> } ) = I ( h ) χ {I h ≤ } , which shows the second equality in (3.8). The last property in (3.7) follows from the definitionof F h , because I ( F h ) = I ( χ {I h> } h ) − I ( χ {I h ≤ } h ) = I ( h ) χ {I h> } − I ( h ) χ {I h ≤ } = |I h | ≥ . Finally, since (cid:82) Ω h (cid:62) = (cid:82) I ( F h ) = (cid:82) Ω F h = I ( F h )(1) = | I h (1) | = | (cid:82) Ω h | = 0, the first equality in(3.7) also holds, and the proof is finished. In this subsection we show the energy-decreasing property of the (cid:62) -transformation. We introducefirst a useful notation. Recall that Ω , δ are as in (3.1) and I := [0 , | Ω | ]. For h ∈ L rad (Ω) let τ : [ δ, → I, τ ( r ) := ω N ( r N − δ N ) (cid:101) h : I → R , (cid:101) h ( s ) := h ◦ τ − ( s ) = h (( ω − N s + δ N ) N ) (3.9)(recall that we use h ( | x | ) = h ( x )). The function (cid:101) h is the one-dimensional equimeasurable versionof h , since, for s ∈ I and r = τ − ( s ) ∈ [ δ, (cid:90) s (cid:101) h = N ω N (cid:90) rδ (cid:101) h ( τ ( ρ )) ρ N − dρ = N ω N (cid:90) rδ h ( ρ ) ρ N − dρ = I h ( r ) . (3.10)17imilarly, observe that |{ h > t }| = (cid:90) Ω χ { h>t } = N ω N (cid:90) δ χ { h>t } ( r ) r N − dr = (cid:90) | Ω | χ { (cid:101) h>t } = |{ (cid:101) h > t }| for t ∈ R , and therefore, (cid:101) h ( s ) = ( h ( τ − ( s ))) = h ( s ) for s ∈ I. (3.11) Proof of Theorem 1.3.
Let f and g as in the statement. By Proposition 3.4, we have that( f (cid:62) , g (cid:62) ) ∈ X and (cid:82) Ω | f | α α + | g | β β = (cid:82) Ω | f (cid:62) | α α + | g (cid:62) | β β , so it suffices to show that (cid:82) Ω gKf ≤ (cid:82) Ω g (cid:62) Kf (cid:62) . Let ( u, v ) := (
Kg, Kf ) and observe that − ( u r r N − ) r r N − = g and − ( v r r N − ) r r N − = f in [ δ, . Then, u r ( r ) = − I g ( r ) N ω N r − N and v r ( r ) = − I f ( r ) N ω N r − N for r ∈ [ δ, . (3.12)Thus, by Proposition 3.4, (cid:90) Ω gKf = N ω N (cid:90) δ − ( u r r N − ) r v = N ω N (cid:90) δ u r v r r N − dr = ( N ω N ) − (cid:90) δ I f ( r ) I g ( r ) r − N dr ≤ ( N ω N ) − (cid:90) δ |I f ( r ) ||I g ( r ) | r − N dr = ( N ω N ) − (cid:90) δ I ( F f )( r ) I ( F g )( r ) r − N dr. (3.13)By (3.10), (cid:90) δ I ( F f )( r ) I ( F g )( r ) r − N dr = (cid:90) δ (cid:90) τ ( r )0 (cid:102) F f ( σ ) dσ (cid:90) τ ( r )0 (cid:102) F g ( σ ) dσ r − N dr = ( N ω N ) − (cid:90) | Ω | (cid:90) s (cid:102) F f ( t ) dt (cid:90) s (cid:102) F g ( σ ) dσ ( ω − N s + δ N ) N − ds. (3.14)Let ϕ ( s ) := ( N ω N ) − ( ω − N s + δ N ) N − ≥
0. Since (cid:102) F f is non-increasing in I and (cid:82) I (cid:102) F f = I ( F f )(1) = 0 (by (3.10) and because averages are preserved by rearrangements), we have that (cid:82) s (cid:102) F f ≥ s ∈ I . Moreover, by Proposition 3.4, (cid:82) s (cid:102) F g = I ( F g )( τ − ( s )) ≥ s ∈ I .Therefore, t (cid:55)→ (cid:90) | Ω | t (cid:90) s (cid:102) F g ( σ ) dσ ϕ ( s ) ds and σ (cid:55)→ (cid:90) | Ω | σ (cid:90) s (cid:102) F f ( t ) dt ϕ ( s ) ds are non-increasing in I . Then, by (the Hardy-Littlewood inequality) Theorem 3.2 and Fubini’stheorem, (cid:90) I (cid:90) s (cid:102) F f ( t ) dt (cid:90) s (cid:102) F g ( σ ) dσ ϕ ( s ) ds = (cid:90) I (cid:102) F f ( t ) (cid:90) | Ω | t (cid:90) s (cid:102) F g ( σ ) dσ ϕ ( s ) ds dt ≤ (cid:90) I (cid:102) F f ( t ) (cid:90) | Ω | t (cid:90) s (cid:102) F g ( σ ) dσ ϕ ( s ) ds dt = (cid:90) I (cid:102) F g ( σ ) (cid:90) | Ω | σ (cid:90) s (cid:102) F f ( t ) dt ϕ ( s ) ds dσ (3.15) ≤ (cid:90) I (cid:102) F g ( σ ) (cid:90) | Ω | σ (cid:90) s (cid:102) F f ( t ) dt ϕ ( s ) ds dσ = (cid:90) I (cid:90) s (cid:102) F g ( σ ) dσ (cid:90) s (cid:102) F f ( t ) dt ϕ ( s ) ds. (3.16)18owever, using (3.10), (3.11) and reasoning as in (3.14), we have (cid:90) I (cid:90) s (cid:102) F g ( σ ) dσ (cid:90) s (cid:102) F f ( t ) dt ϕ ( s ) ds = (cid:90) I (cid:90) s ( F f ) ( σ ) dσ (cid:90) s ( F g ) ( σ ) dσ ϕ ( s ) ds = (cid:90) I (cid:90) s (cid:102) f (cid:62) ( σ ) dσ (cid:90) s (cid:102) g (cid:62) ( σ ) dσ ϕ ( s ) ds = (cid:90) δ I ( f (cid:62) )( r ) I ( g (cid:62) )( r ) r − N dr. Therefore, arguing as in (3.13), we obtain that (cid:90) Ω gKf ≤ ( N ω N ) − (cid:90) δ I ( f (cid:62) )( r ) I ( g (cid:62) )( r ) r − N dr = (cid:90) Ω g (cid:62) Kf (cid:62) , (3.17)and (1.11) follows.We now show that if f and g are nontrivial and φ ( f (cid:62) , g (cid:62) ) = φ ( f, g ), then f and g aremonotone in the radial variable. Observe that, since f is nontrivial, (cid:102) F f is non-increasing,and (cid:82) I (cid:102) F f = I ( F h )(1) = 0, then (cid:82) s (cid:102) F f > s ∈ (0 , | Ω | ). But then the function σ (cid:55)→ (cid:82) | Ω | σ (cid:82) s (cid:102) F f ( t ) dt ϕ ( s ) ds is a strictly decreasing continuous function in I and, in virtue ofLemma 3.3, equality in (3.16) may only hold if (cid:102) F g = (cid:102) F g a.e. in I , which, by (3.9), implies that F g coincides a.e. with a radially monotone function. Then g must also be radially monotone,because | F g | = | g | in Ω, g is a nontrivial continuous function, and (cid:82) Ω g = 0. Arguing similarlyusing (3.15) we conclude that f is monotone in the radial variable as well. As a consequence, if( u, v ) := ( K p g, K q f ), then u and v are strictly monotone in the radial variable by (3.12) and thefact that either I f > I g > I f < I g < δ, Remark 3.5.
To motivate the definition of f (cid:62) , observe that the flipping F f is very naturalin virtue of (3.13), but further insight on F f can be gained by observing that, by (3.12), the(Neumann radial) solution of − ∆ v = F f in Ω satisfies that v r ( r ) = − ( N ω N ) − r − N I ( F f )( r ) ≤ , v is decreasing in the radial variable. By (3.13), this is indeed a very simple wayto guarantee that there is at least one minimizer which is monotone in the radial variable;but the flipping F f is too simple to extract any further information. To show that all leastenergy solutions ( u, v ) := ( K p g, K q f ) are strictly monotone decreasing, we need to combine theflipping F with the decreasing rearrangement Remark 3.6.
In general, it is not true that φ ( f ∗ , g ∗ ) ≤ φ ( f, g ), where ∗ denotes the Schwarzsymmetrization and ( f, g ) is as in Theorem 1.3. A first observation is that f ∗ is defined in Ω ∗ (a ball with the same measure as Ω), and this would be a first complication when consideringannular domains. But even if Ω is a unitary ball centered at zero B and we consider the scalarproblem ( i.e. , p = q , see Lemma 2.6), we exhibit next a continuous radial function f : B → R with (cid:82) B f = 0 such that φ ( f ∗ , f ∗ ) > φ ( f, f ). Let N = 5, ε = , and f ( x ) := − ( ω N | x | N + ε ) − + k ,where k := 2 ω − N (( ω N + ε ) − ε ) is such that (cid:82) B f = 0. In particular, (cid:101) f ( s ) = − ( s + ε ) − + k for s ∈ I = [0 , ω N ] = [0 , π ]. Observe that (cid:101) f is a strictly increasing function; therefore, by (3.11), f ∗ ( x ) = f ( ω N | x | N ) = (cid:101) f ( ω N | x | N ) = (cid:101) f ( ω N − ω N | x | N ). We have that (cid:90) I (cid:18)(cid:90) s (cid:101) f (cid:19) ( ω − N s ) N − ds ≈ . > . ≈ (cid:90) I (cid:18)(cid:90) s f (cid:19) ( ω − N s ) N − ds. f = (cid:101) f ∗ in I , we may use the techniques and notationfrom the proof of Theorem 1.3 and we find that (cid:90) B f Kf = ( N ω N ) − (cid:90) I (cid:18)(cid:90) s (cid:101) f (cid:19) ϕ ( s ) ds > ( N ω N ) − (cid:90) I (cid:18)(cid:90) s (cid:102) f ∗ (cid:19) ϕ ( s ) ds = (cid:90) B f ∗ Kf ∗ , where ϕ ( s ) := ( N ω N ) − ( ω − N s ) N − for s ∈ I . Therefore φ ( f ∗ , f ∗ ) > φ ( f, f ), because (cid:107) f ∗ (cid:107) α = (cid:107) f (cid:107) α by (3.2). Remark 3.7.
The Neumann b.c. are used at (3.12) (if Ω is an annulus) and in the integrationby parts performed in (3.13). The Neumann b.c. is also the reason to consider zero-averagefunctions.
Theorem 3.8. (Sublinear case) Let ( p, q ) satisfy (1.19) and Ω and δ be as in (3.1) . The set ofminimizers of φ in X rad is nonempty. If ( f, g ) ∈ X rad is such that φ ( f, g ) = inf X rad φ , then f and g are monotone in the radial variable and, if ( u, v ) := ( K p g, K q f ) , then u r v r > in ( δ, ;that is, u and v are both strictly monotone increasing or strictly monotone decreasing in theradial variable.Proof. Arguing as in Lemma 2.8 we have that the set of minimizers of φ in X rad is nonemptyand contains only nontrivial functions. By Proposition 2.4 we have that all such minimizers( f, g ) are continuous up to the boundary ∂ Ω and ( u, v ) := ( K p g, K q f ) solves (1.1) pointwise, inparticular ( f, g ) = ( | u | p − u, | v | q − v ). By Theorem 1.3, u and v are strictly monotone. To showthat u r v r > δ, u is increasing; then u r > δ, v r = − ( N ω N ) − I f ( r ) r − N = − ( N ω N ) − I ( | u | p − u )( r ) r − N > δ, v is also strictly increasing. This ends the proof. Theorem 3.9. (Superlinear case) Let ( p, q ) satisfy (1.20) and Ω and δ be as in (3.1) . The setof minimizers of φ in N rad is nonempty. If ( f, g ) ∈ N rad is such that φ ( f, g ) = inf N rad φ , then f and g are monotone in the radial variable and, if ( u, v ) := ( K p g, K q f ) , then u r v r > in ( δ, .Proof. Arguing precisely as in Lemma 2.10, there exists at least one minimizer ( f, g ) of φ in N rad .Since the L s norms are preserved by the (cid:62) -transformation, and (3.17) holds, then reasoning asin part 3) of the proof of Lemma 2.10 we have that ( f (cid:62) , g (cid:62) ) is also a minimizer of φ in N rad .The rest of the proof follows as in the proof of Theorem 3.8.Since the proof of Theorem 3.8 is based on one-dimensional techniques, one can also applythem to a least energy solution in the case N = 1 to obtain monotonicity properties. Corollary 3.10.
Let
Ω = ( − , , ( f, g ) as in (1.8) , and ( u, v ) := ( K p g, K q f ) . Then u r v r > in ( − , , that is, u and v are both strictly monotone increasing or strictly monotone decreasingin the radial variable.Proof. For N = 1 and Ω = ( − , I : L ∞ (Ω) → C (Ω) and F : C (Ω) → L ∞ (Ω) be given by I h ( x ) = (cid:90) x − h ( ρ ) dρ and F h := ( χ {I h> } − χ {I h ≤ } ) h. Let τ : [ − , → [0 , x (cid:55)→ τ ( x ) = x + 1, and for h ∈ C (Ω), the (cid:62) -transformation of h is givenby h (cid:62) ∈ L ∞ (Ω) , h (cid:62) ( x ) := ( F h ) ( x + 1) . The proof now follows by replacing I , F , τ , and (cid:62) instead of I , F , τ , and (cid:62) in the proofs ofProposition 3.4 and Theorems 3.8 and 3.9 using δ := − Symmetry and symmetry breaking of minimizers
This subsection is devoted to the following result.
Theorem 4.1.
Let N ≥ , Ω as in (1.9) , (1.2) hold, and ( f, g ) satisfy (1.8) , then f and g arenot radial. The proof of this result is long, and we split it in several lemmas. The argument goes bycontradiction and is also based on the results from Section 3, i.e. , that if f and g are radial, thenboth functions are either strictly increasing or strictly decreasing. This allows to construct adirection ( ϕ, ψ ) along which the second variation of φ at ( f, g ) is strictly negative, contradictingthe minimality property of ( f, g ). In the following, Ω ⊂ R N is either a ball or an annulus as in(3.1). We begin with an auxiliary lemma. Lemma 4.2.
Let U ⊂ R N be a domain of class C , and W ( U ) := { w ∈ C ( U ) : ∇ w ( x ) (cid:54) = 0 whenever x ∈ U satisfies w ( x ) = 0 } , (4.1) which is an open subset of C ( U ) . For s > the following holds.(1) There is δ > depending only on s such that the map γ : W ( U ) → L δ ( U ) , γ ( w ) := | w | s − is well-defined and continuous.(2) If w ∈ W ( U ) , then ( | w | s − w ) x i = s | w | s − w x i , i = 1 , . . . , N , and (cid:90) U | w | s − w ∂ x i h = s (cid:90) U | w | s − w x i h for all h ∈ C ∞ c ( U ) . The proof of Lemma 4.2 follows from (the proof of) [25, Proposition 2.3], where only the case s ∈ (0 ,
1) is considered, but the case s ≥ φ is nonnegative along some directions. Weuse X rad to denote the subspace of radially symmetric functions in X . Lemma 4.3.
Let pq < and ( f, g ) ∈ X rad be a global minimizer of φ in X , ( u, v ) := ( K p g, K q f ) ,and let ( ϕ, ψ ) ∈ [ C (Ω)] ∩ X , with supp( ϕ ) ⊂ Ω \ f − (0) and supp( ψ ) ⊂ Ω \ g − (0) . Then themap s (cid:55)→ φ ( f + sϕ, g + sψ ) belongs to C ∞ ( R ) and lim s → d ds φ ( f + sϕ, g + sψ ) = (cid:90) Ω p | u | − p ϕ + 1 q | v | − q ψ − ϕKψ − ψKϕ dx ≥ . (4.2) Proof.
For s ∈ R , we write φ ( f + sϕ, g + sψ ) = Ψ( s ) − T ( s ), with ψ ( s ) = (cid:90) Ω | f + sϕ | α α + | g + sψ | β β dx and T ( s ) = (cid:90) Ω ( f + sϕ ) K ( g + sψ ) = (cid:90) Ω f Kg + s (cid:90) Ω ϕKg + f Kψ dx + s (cid:90) Ω ϕKψ. Then T ∈ C ∞ ( R ) and, by integration by parts, T (cid:48)(cid:48) (0) = 2 (cid:90) Ω ϕKψ dx = (cid:90) Ω ϕKψ + ψKϕ dx.
21s for Ψ, since α = p +1 p > β = q +1 q >
1, it is standard to show thatΨ (cid:48) ( s ) = (cid:90) Ω | f + sϕ | α − ( f + sϕ ) ϕ + (cid:90) Ω | g + sψ | β − ( g + sψ ) ψ. Since u ∈ C (Ω) and U := supp( ϕ ) ⊂ Ω \ f − (0), then f = | u | p − u is bounded away from 0 on U ,and | f | α − ∈ C ( U ). Analogously, | g | β − ∈ C ( V ) with V := supp( ψ ). Therefore ψ ∈ C ∞ ( R )and ψ (cid:48)(cid:48) (0) = lim s → s ( (cid:90) U (cid:0) | f + sϕ | α − ( f + sϕ ) − | f | α − f (cid:1) ϕ + (cid:90) V (cid:0) | g + gϕ | α − ( g + sψ ) − | g | β − g (cid:1) ψ )= (cid:90) U ( α − | f | α − ϕ dx + (cid:90) V ( β − | g | β − ψ , as required.Let N rad be the subset of radial functions in N . We say that a function ϕ : Ω → R isantisymmetric with respect to x if ϕ ( x , x (cid:48) ) = − ϕ ( − x , x (cid:48) ) for all ( x , x (cid:48) ) ∈ Ω. Lemma 4.4.
Let pq > and ( f, g ) ∈ N rad be a minimizer of φ in N rad , ( u, v ) := ( K p g, K q f ) .Let ( ϕ, ψ ) ∈ [ C (Ω)] ∩ X be antisymmetric with respect to x and such that supp( ϕ ) ⊂ Ω \ f − (0) and supp( ψ ) ⊂ Ω \ g − (0) . Then the map s (cid:55)→ φ ( f + sϕ, g + sψ ) belongs to C ∞ ( R ) and (4.2) holds.Proof. By Lemma 4.3 the map s (cid:55)→ φ ( f + sϕ, g + sψ ) is C at s = 0. We now argue asin [25, Theorem 4.2]. Assume, by contradiction, that there is ( ϕ, ψ ) ∈ [ C (Ω)] ∩ X antisymmetricwith respect to x satisfying supp( ϕ ) ⊂ Ω \ f − (0), supp( ψ ) ⊂ Ω \ g − (0), and such that d ds φ ( f + sϕ, g + sψ ) < − κ for s ∈ ( − κ , κ ) and some κ, κ >
0. Let γ and γ as in (2.13). Using thefact that ( f, g ) is radially symmetric and ( ϕ, ψ ) is antisymmetric with respect to x , we obtainthat φ (cid:48) ( c γ f, c γ g )( c γ ϕ, c γ ψ ) = 0 for c ∈ [ , φ ( c γ ( f + sϕ ) , c γ ( g + sψ )) ≤ φ ( c γ f, c γ g ) − κs + o ( s ) for s ∈ ( − κ , κ ) , where o ( s ) is a remainder term uniform in c ∈ [ , s ∈ ( − κ , κ ) sufficiently small, let t = t ( f + sϕ, g + sψ ) > t ∈ [ , f, g ) ∈ N , φ ( t γ ( f + sϕ ) , t γ ( g + sψ )) = sup c ∈ [ , φ ( c γ ( f + sϕ ) , c γ ( g + sψ )) ≤ sup c ∈ [ , φ ( c γ f, c γ g ) − κs = φ ( f, g ) − κs < φ ( f, g ) , which yields a contradiction to the minimality of ( f, g ), because ( t γ ( f + sϕ ) , t γ ( g + sψ )) ∈ N ,and the claim follows.In the case of least energy radial solutions, Lemma 4.3 and Lemma 4.4 allow us to concludethe following. Let Ω( e ) := { x ∈ Ω : x > } . Lemma 4.5.
Let ( f, g ) ∈ X rad satisfy (1.10) and ( u, v ) := ( K p g, K q f ) . If u is increasing in theradial variable and ( ¯ f , ¯ g ) := ( p | u | p − u x , q | v | q − v x ) , then K ¯ f ≥ and K ¯ g ≥ in Ω( e ) (4.3) and ∞ > (cid:90) Ω( e ) p | u | p − u x ( u x − K ¯ g ) + q | v | q − v x ( v x − K ¯ f ) dx ≥ , (4.4)22 roof. Note that ( u, v ) := ( K p g, K q f ) ∈ [ C ,ε (Ω)] \ { (0 , } is a radial classical solution of (1.1),by Proposition 2.4. By Theorem 1.2 we know that u r v r > δ, δ = inf x ∈ Ω | x | . Byassumption u r ≥ δ, u and v are strictly monotone increasing in the radial variable.In particular, ¯ f := p | u | p − u x ≥ g := q | v | q − v x ≥ e ) . Moreover, since u and v are sign-changing, there exist r , r ∈ ( δ,
1) such that u − (0) = f − (0) = { x ∈ Ω : | x | = r } and v − (0) = g − (0) = { x ∈ Ω : | x | = r } , i.e., the nodal sets are two spheres contained in Ω and u, v ∈ W (Ω), with W (Ω) as defined in(4.1). To control these nodal lines we use the following cutoff functions: for ε, r ≥
0, let ρ rε be asmooth radial function in R N such that0 ≤ ρ rε ≤ , ρ rε ( x ) = 0 if | | x | − r | < ε, and ρ rε ( x ) = 1 if | | x | − r | > ε, and let ( ¯ f ε , ¯ g ε ) := ( ¯ f ρ r ε , ¯ gρ r ε ) for ε ≥
0; note that ( ¯ f ε , ¯ g ε ) ∈ [ C (Ω)] for ε > f ε , ¯ g ε ) isantisymmetric in Ω with respect to x for all ε ≥
0, because ( u, v ) is radially symmetric in Ω.Moreover,( K ¯ g ε , K ¯ f ε ) ∈ [ C (Ω) ∩ C (Ω)] for ε > , ( K ¯ g , K ¯ f ) = ( K ¯ g, K ¯ f ) ∈ [ W ,t (Ω)] for some t > K ¯ g ε , K ¯ f ε ) is also antisymmetric in Ω with respect to x for all ε ≥
0, by uniqueness. In particular, K ¯ f ε = K ¯ g ε = 0 on ∂ Ω( e ) ∩ { x = 0 } and K ¯ g ε ≥ e ) and K ¯ f ε ≥ e ) , (4.5)by the maximum principle and Hopf’s boundary point lemma. Since ( ¯ f ε , ¯ g ε ) → ( ¯ f , ¯ g ) in L t (Ω( e ))as ε →
0, then ( K ¯ f ε , K ¯ g ε ) → ( K ¯ f , K ¯ g ) in W ,t (Ω( e )) as ε → , (4.6)and therefore (4.5) implies (4.3). Furthermore, since the product of two antisymmetric functionsis symmetric, we have that2 (cid:90) Ω( e ) ¯ f ε ( u x ρ r ε − K ¯ g ε ) + ¯ g ε ( v x ρ r ε − K ¯ f ε ) dx = (cid:90) Ω ¯ f ε ( u x ρ r ε − K ¯ g ε ) + ¯ g ε ( v x ρ r ε − K ¯ f ε ) dx ≥ , (4.7)by applying Lemma 4.3 in the sublinear case (1.19) or Lemma 4.4 in the superlinear case (1.20)with ( ϕ, ψ ) := ( ¯ f ε , ¯ g ε ) for small ε >
0. The claim (4.4) follows from Lebesgue’s dominatedconvergence theorem once we show the existence of a suitable majorant. Indeed, let ( ξ ε , ζ ε ) :=( K ¯ g ε , K ¯ f ε ) for ε ≥
0, then, by (4.6),¯ f ε ( u x ρ r ε − ξ ε ) → | u | p − u x ( u x − ξ ) a.e. in Ω( e ) as ε → . Moreover, we have that − ∆( ξ ε − ξ ) = ¯ g ε − ¯ g = q | v | q − v x ( ρ r ε − ≤ e ), with ∂ ν ( ξ ε − ξ ) =0 on ∂ Ω( e ) \{ x = 0 } and ξ ε − ξ = 0 on ∂ Ω( e ) ∩ { x = 0 } . Thus (testing the equation with( ξ ε − ξ ) + and integrating by parts), 0 < ξ ε ≤ ξ in Ω( e ) for ε small, and (cid:12)(cid:12) ¯ f ε ( u x ρ r ε − ξ ε ) (cid:12)(cid:12) ≤ | u | p − u x + | u | p − u x ξ ε ≤ | u | p − u x + | u | p − u x ξ . | u | p − u x ∈ L (Ω), because | u | p − ∈ L t (Ω) ⊂ L (Ω) and u ∈ C (Ω). It remains toshow that | u | p − u x ξ ∈ L (Ω) (4.8)Since − ∆( u x − ξ ) = 0 in Ω in the strong sense, interior elliptic regularity (see, e.g. , [18,Theorem 9.19] ) implies that u x − ξ ∈ C ∞ (Ω); therefore ξ ∈ C (Ω) ∩ W ,t (Ω), because u x ∈ C (Ω). This directly implies (4.8) for p ≥
1. If p ∈ (0 , γ > A := { x ∈ Ω : r − γ < | x | < r + γ } ⊂ Ω. Then (cid:90) Ω( e ) | u | p − | u x ξ | = (cid:90) Ω( e ) \ A | u | p − | u x ξ | + (cid:90) A | u | p − | u x ξ |≤ ( min Ω( e ) \ A | u | ) p − (cid:107) u x (cid:107) L ∞ (Ω) (cid:90) Ω( e ) | ξ | + (cid:107) u x ξ (cid:107) L ∞ ( A ) (cid:90) Ω( e ) | u | p − < ∞ , and (4.8) also follows. A majorant for the term ¯ g ε ( v x ρ r ε − K ¯ f ε ) in (4.7) can be obtainedsimilarly, and this ends the proof.The following lemma shows that an antisymmetric Neumann solution dominates the corre-sponding Dirichlet solution in a half radial domain Ω( e ). Lemma 4.6.
Let t > and h ∈ L t (Ω) \{ } be an antisymmetric function in Ω with respect to x and let w N := Kh ∈ W ,t (Ω) , that is, w N is the unique strong solution of − ∆ w N = h in Ω , ∂ ν w N = 0 on ∂ Ω , and (cid:90) Ω w N = 0 . Moreover, let w D ∈ W ,t (Ω) ∩ C (Ω) be a strong solution of − ∆ w D = h in Ω with w D = 0 on ∂ Ω . If h ≥ and w N ≥ in Ω( e ) then w D < w N in Ω( e ) .Proof. By uniqueness of solutions we have that w D ∈ W ,t (Ω) ∩ C (Ω) and w N := Kh ∈ W ,t (Ω)are antisymmetric functions in Ω with respect to x and therefore, since w N ≥ e ) byassumption, w D − w N = 0 on ∂ Ω( e ) ∩ { x = 0 } and w D − w N ≤ ∂ Ω( e ) ∩ { x > } . (4.9)Observe that w D (cid:54)≡ w N in Ω( e ), because otherwise w D = w N ≥ e ) with ∂ ν w D = w D = 0 on ∂ Ω( e ) ∩ { x > } , which contradicts Hopf’s boundary point lemma. Then, since − ∆( w D − w N ) = 0 in Ω, w D (cid:54)≡ w N in Ω( e ), and (4.9) holds, the maximum principle yields that w D < w N in Ω( e ). Proof of Theorem 4.1.
Let Ω ⊂ R N as in (3.1), ( f, g ) as in the statement, ( u, v ) := ( K p g, K q f ) ∈ [ C ,ε ( B )] , and ( ¯ f , ¯ g ) := ( p | u | p − u x , q | v | q − v x ). We argue by contradiction. Assume withoutloss of generality that f is radial. Using the relations u = | f | p f, v = K q f , and g = | v | q − v, weobtain that also u , v , and g must be radially symmetric. By Lemma 4.2, there is some t > | u | p − , | v | q − ∈ L t (Ω) and ∂ x ( | v | q − v ) = q | v | q − v x ∈ L t (Ω). Therefore, we have that − ∆ u = | v | q − v ∈ W ,t (Ω) and (interior) elliptic regularity (see for instance [18, Theorem 9.19])yields that u ∈ W ,t loc (Ω). Thus, we may interchange derivatives, and ( − ∆ u ) x = − ∆( u x ) in Ω.Arguing analogously for − ∆ v we conclude that ( u x , v x ) ∈ [ W ,t (Ω) ∩ C ,ε (Ω)] is the uniquestrong solution of the Dirichlet problem − ∆ u x = q | v | q − v x , − ∆ v x = p | u | p − u x in Ω with u x = v x = 0 on ∂ Ω , u and v are radially symmetric and ∂ ν u = ∂ ν v = 0 on ∂ Ω. By Theorem 1.2, we may assume that u and v are strictly increasing inthe radial variable (the other case follows similarly). Then u x and v x are nonnegative in Ω( e )and, by Lemmas 4.5 and 4.6,0 ≤ (cid:90) Ω( e ) p | u | p − u x ( u x − K ¯ g ) + q | v | q − v x ( v x − K ¯ f ) dx < , a contradiction. In this section we show that least energy solutions are foliated Schwarz symmetric whenever thedomain Ω is a ball or an annulus centered at zero in R N in dimension N ≥
2. For N = 1, seeCorollary 3.10. We introduce first some notation. Let S N − = { x ∈ R N : | x | = 1 } be the unitsphere and fix e ∈ S N − . We consider the halfspace H ( e ) := { x ∈ R N : x · e > } and the halfdomain Ω( e ) := { x ∈ Ω : x · e > } . The composition of a function w : Ω → R with a reflection with respect to ∂H ( e ) is denotedby w e , that is, w e : Ω → R is given by w e ( x ) := w ( x − x · e ) e ) . The polarization u H of u : Ω → R with respect to a hyperplane H = H ( e ) is given by u H := (cid:40) max { u, u e } in Ω( e ) , min { u, u e } in Ω \ Ω( e ) . Following [34], we say that u ∈ C (Ω) is foliated Schwarz symmetric with respect to some unitvector p ∈ S N − if u is axially symmetric with respect to the axis R p and nonincreasing in thepolar angle θ := arccos( x | x | · p ) ∈ [0 , π ] . We use the following characterization of foliated Schwarzsymmetry given in [32]. We remark that these kind of characterizations appeared for the firsttime in [10], see also [35] for a survey on symmetry via reflection methods.
Lemma 4.7 (Particular case of Proposition 3.2 in [32]) . There is p ∈ S N − such that u, v ∈ C (Ω) are foliated Schwarz symmetric with respect to p if and only if for every e ∈ S N − either u ≥ u e , v ≥ v e in Ω( e ) or u ≤ u e , v ≤ v e in Ω( e ) . (4.10)The main result of this section is the following. For similar results under Dirichlet boundaryconditions, we refer to [7, Theorem 1.3] and [8, Theorem 1.2]. Theorem 4.8 (Sublinear case) . Let Ω be either a ball or an annulus centered at the originof R N , N ≥ . Let ( p, q ) satisfy (1.19) and ( f, g ) ∈ X be a global minimizer of φ in X and ( u, v ) := ( K p g, K q f ) . There is p ∈ S N − such that u and v are foliated Schwarz symmetric withrespect to p .Proof. Let ( f, g ) and ( u, v ) as in the statement and fix a hyperplane H = H ( e ) for some | e | = 1.By Proposition 2.4, ( u, v ) ∈ [ C ,µ (Ω)] for some µ ∈ (0 , u, v ) solves (1.1) pointwise, and( f, g ) = ( − ∆ v, − ∆ u ) ∈ [ C µ (Ω)] ; thus ( f H , g H ) ∈ [ L ∞ (Ω)] and ( (cid:101) u, (cid:101) v ) := ( K p ( g H ) , K q ( f H )) ∈ [ W ,N (Ω) ∩ C (Ω)] , by Sobolev embeddings. 25et V := v e + v − (cid:101) v − (cid:101) v e , then using the definition of f H we have that − ∆ V = f − f H − ( f H ) e + f e = 0 in Ω , ∂ ν V = 0 on ∂ Ω . testing this equation with V and integrating by parts we obtain that V = k for some k ∈ R .Then v e + v = (cid:101) v e + (cid:101) v + k in Ω (4.11)Let Γ := { x ∈ ∂ Ω( e ) : x · e = 0 } , Γ := { x ∈ ∂ Ω( e ) : x · e > } ,w := (cid:101) v − v + k/
2, and w := (cid:101) v − v e + k/
2. Since v = v e and (cid:101) v = (cid:101) v e on Γ we have that w = w = 0 on Γ , by (4.11), and ∂ ν w = ∂ ν w = 0 on Γ . Furthermore, − ∆ w = f H − f ≥ e ) and − ∆ w = f H − f e ≥ e ) , which implies by the maximum principle and Hopf’s Lemma that w ≥ w ≥ e ).Therefore, using that (cid:101) v e = v e + v − (cid:101) v − k (by (4.11)) and g He = g e + g − g H (by definition of g H ), (cid:90) Ω gKf − g H Kf H dx = (cid:90) Ω gv − g H (cid:101) v = (cid:90) Ω( e ) gv + g e v e − g H (cid:101) v − ( g H ) e (cid:101) v e dx = (cid:90) Ω( e ) gv + g e v e − g H (cid:101) v − ( g e + g − g H )( v e + v − (cid:101) v − k ) dx = (cid:90) Ω( e ) ( g e − g H ) w + ( g − g H ) w + k g e + g ) dx ≤ , (4.12)because (cid:82) Ω g = 0.To show that u and v are foliated Schwarz symmetric with respect to the same vector p ∈ S N − we use Lemma 4.7 and argue by contradiction. Assume that (4.10) does not hold. Then, withoutloss of generality, there are e ∈ S N − and the corresponding halfspace H = H ( e ) such that v (cid:54) = v H in B ( e ) and either v e (cid:54) = v H in Ω( e ) or u e (cid:54) = u H in Ω( e ). Since t (cid:55)→ h ( t ) := | t | s t is a strictlymonotone increasing function in R for s > −
1, this implies that f = h ( v ) (cid:54) = h ( v ) H = f H and either f e (cid:54) = f H or g e (cid:54) = g H in Ω( e ) . As a consequence, w > (cid:54) = g − g H ≤ B ( e ) and either w > B ( e ) or 0 (cid:54) = g e − g H ≤ B ( e ). In any case, (4.12) implies that (cid:90) Ω f Kg < (cid:90) Ω f H Kg H . (4.13)But then, since L s norms are preserved under polarizations, we obtain that φ ( f, g ) > φ ( f H , g H ),a contradiction to the minimality of ( f, g ). Therefore (4.10) holds and the theorem follows fromLemma 4.7. Theorem 4.9 (Superlinear case) . Let Ω be either a ball or an annulus centered at the originof R N , N ≥ . Let ( p, q ) satisfy (1.20) and ( f, g ) ∈ X be a minimizer of φ in N and ( u, v ) :=( K p g, K q f ) . There is p ∈ S N − such that u and v are foliated Schwarz symmetric with respectto p . roof. Arguing by contradiction as in the proof of Theorem 4.8, we obtain (4.13). Since ( f, g ) ∈N and L s -norms are preserved under polarizations we have, by (2.14), (cid:90) Ω γ | f H | α + γ | g H | β dx < (cid:90) Ω f H Kg H and (cid:82) Ω f H Kg H >
0, then by Lemma 2.9 there exists 0 < t < t γ f H , t γ g H ) ∈ N .This gives a contradiction, because, by (2.14),inf N φ ≤ φ ( t γ f H , t γ g H ) = t γ (1 − γ ) (cid:90) Ω | f H | α α + | g H | β β dx< (1 − γ ) (cid:90) Ω | f | α α + | g | β β dx = φ ( f, g ) . In this section, Ω is again a general smooth bounded domain. We prove that if ( u, v ) is a solutionof (1.1) associated to a minimizer of φ , then the nodal sets u − (0) := { x ∈ Ω : u (0) = 0 } and v − (0) := { x ∈ Ω : v ( x ) = 0 } have zero Lebesgue measure. We do it by extendingthe results and techniques from [25, Section 3] to the setting of Hamilitonian elliptic systemsand to the dual method framework. Recall that if ( f, g ) ∈ X is a critical point of φ then( u, v ) := ( K p g, K q f ) ∈ [ C ,ε (Ω)] is a classical solution of (1.1). Our main result is the following. Theorem 5.1.
Let ( f, g ) ∈ X be a critical point of φ in X and ( u, v ) := ( K p g, K q f ) , then u − (0) = v − (0) a.e. Moreover, if p and q satisfy the sublinear condition (1.19) and ( f, g ) ∈ X is a global minimizer of φ in X , then | u − (0) | = | v − (0) | = 0 . To show Theorem 5.1 we rely on the following preliminary results.
Lemma 5.2 (Lemma 3.1 in [25]) . Let γ > and f : (0 , ∞ ) → [0 , ∞ ) be such that f is boundedin [ ε, ∞ ) for every ε > and lim r → r γ f ( r ) = 0 . Then for every r > there is s > such that f ( s ) ≥ f ( r ) and f ( t ) ≤ γ f ( s ) for all t ∈ [ s , s ] . We now characterize the decay of any solution ( u, v ) of (1.1) close to a common zero.
Proposition 5.3.
Let ( u, v ) ∈ [ C (Ω)] be a solution of (1.1) with p, q > , pq < . If x ∈ Ω is a point of density one for the set u − (0) ∩ v − (0) then | u ( x ) | p +1 + | v ( x ) | q +1 = o ( | x − x | γ ) ,where γ = 2( p + 1)( q + 1)1 − pq = 2 (cid:18) − α − β (cid:19) − . (5.1) In other words,if lim r → | u − (0) ∩ v − (0) ∩ B r ( x ) || B r ( x ) | = 1 then lim x → x | u ( x ) | p +1 + | v ( x ) | q +1 | x − x | γ = 0 . Proof.
Without loss of generality we assume that x = 0 and we set u ≡ v ≡ R N \ Ω. Let f : (0 , ∞ ) → [0 , ∞ ) be given by f ( r ) := r − γ sup | x | = r ( | u ( x ) | p +1 + | v ( x ) | q +1 ) . tep 1. We show first that f ∈ L ∞ (0 , ∞ ). Indeed, assume by contradiction that there is asequence ( r n ) n ∈ N such that f ( r n ) → ∞ as n → ∞ . Then, by Lemma 5.2, there is ( s n ) n ∈ N suchthat f ( s n ) ≥ f ( r n ) and f ( t ) ≤ γ f ( s n ) for all t ∈ [ s n , s n ] and n ∈ N . Then f ( s n ) → ∞ as n → ∞ and s n →
0, by the definition of f . Working if necessary with a subsequence, we mayassume that B s n ⊂ Ω for all n ∈ N . Set Ω := B \ B and let u n , v n : Ω → R be given by u n ( x ) = u ( s n x ) f ( s n ) p +1 s γp +1 n and v n ( x ) = v ( s n x ) f ( s n ) q +1 s γq +1 n . (5.2)By (1.1), and since 2 − γp +1 + q γq +1 = 2 − γq +1 + p γp +1 = 0 we have that − ∆ u n = A n | v n | q − v n and − ∆ v n = A n | u n | p − u n in Ω , (5.3)where A n = f ( s n ) − γ = f ( s n ) pq − p +1)( q +1) → n → ∞ . (5.4)Observe next that | u n ( x ) | p +1 + | v n ( x ) | q +1 = | x | γ | u ( s n x ) | p +1 + | v ( s n x ) | q +1 ( | x | s n ) γ f ( s n ) ≤ γ f ( s n | x | ) f ( s n ) ≤ γ (5.5)for all x ∈ Ω and n ∈ N . By (5.3), (5.4), (5.5), and interior elliptic regularity, there aresubsequences u n → u ∗ , v n → v ∗ in C loc (Ω ). Furthermore, by definition of f and the regularityof u, v , there is ( x n ) n ∈ N ⊂ S N − such that | u n ( x n ) | p +1 + | v n ( x n ) | q +1 = 1 for all n ∈ N . Thus, upto a subsequence, x n → x ∗ ∈ S N − with | u ∗ ( x ∗ ) | p +1 + | v ∗ ( x ∗ ) | q +1 = 1 . (5.6)However, since 0 is a point of density one for u − (0) ∩ v − (0), we have that |{ x ∈ Ω : u n ( x ) (cid:54) = 0 }|| B | ≤ |{ x ∈ B s n : u ( x ) (cid:54) = 0 }|| B s n | → n → ∞ , which implies that u ∗ ≡ . Analogously, we obtain that v ∗ ≡ . This contradicts(5.6) and therefore f ∈ L ∞ (0 , ∞ ). Step 2.
Now, it suffices to show that lim r → f ( r ) = 0. We argue again by contradiction: assumethere is a sequence r n → n → ∞ such that f ( r n ) ≥ ε for all n ∈ N and for some ε > B r n ⊂ Ω. Since f is bounded (by Step 1),the rescaled functions (cid:101) u n , (cid:101) v n : Ω → R given by (cid:101) u n ( x ) = u ( r n x ) r γp +1 n and (cid:101) v n ( x ) = v ( r n x ) r γq +1 n (5.7)are uniformly bounded. Moreover, − ∆ (cid:101) u n = | (cid:101) v n | q − (cid:101) v n and − ∆ (cid:101) v n = | (cid:101) u n | p − (cid:101) u n in Ω , (5.8)by (1.1), and there is ( (cid:101) x n ) n ∈ N ⊂ S N − such that | (cid:101) u n ( x n ) | p +1 + | (cid:101) v n ( x n ) | q +1 = f ( r n ) ≥ ε forall n ∈ N . But arguing as before, we obtain subsequences (cid:101) u n → (cid:101) u , (cid:101) v n → (cid:101) v in C loc (Ω ), and (cid:101) x n → (cid:101) x ∈ S N − such that | (cid:101) u ( (cid:101) x ) | + | (cid:101) v ( (cid:101) x ) | ≥ ε . But (5.4) with (cid:101) u n instead of u n yields that (cid:101) u ≡ (cid:101) v ≡
0, a contradiction. Therefore lim r → f ( r ) = 0, and the proofis finished. 28e now use Proposition 5.3 to construct directions along which the energy φ decreases. Proposition 5.4.
Let ( f, g ) ∈ X be a critical point of φ , ( u, v ) := ( K p g, K q f ) , and assume that (1.19) holds. If | u − (0) ∩ v − (0) | > , then there is ( ϕ, ψ ) ∈ X such that φ ( f, g ) > φ ( f + ϕ, g + ψ ) .Proof. If | u − (0) ∩ v − (0) | > x of u − (0) ∩ v − (0). Without loss of generality weassume that x = 0. Since ( u, v ) := ( K p g, K q f ) is a classical solution of (1.1) by Proposition 2.4,we obtain by Proposition 5.3 that | g | β = | ∆ u | β = | v | q +1 = o ( | x | γ ) and | f | α = | ∆ v | α = | u | p +1 = o ( | x | γ ) , (5.9)as | x | →
0, where γ is as in (5.1). Let ζ ∈ C ∞ c ( R N ) \{ } such that supp ζ ⊂ Ω, (cid:82) Ω ζ = 0, and fix t > c := t αβα + β (cid:90) Ω | ζ | α α + | ζ | β β dx − t (cid:107) ζ (cid:107) L (Ω) (cid:107)∇ ζ (cid:107) L (Ω) < t exists, since αβ/ ( α + β ) >
1, which is equivalent to pq < r > r := r Ω ⊂ Ω and let ϕ r , ψ r : Ω → R be given by ϕ r ( x ) := r γα t βα + β ζ ( xr ) , ψ r ( x ) := r γβ t αα + β ζ ( xr )Notice that ϕ r ≡ ψ r ≡ R N \ Ω r and note that w := K ( ζ ( xr )) ∈ C (Ω) ∩ C (Ω) solvesclassically − ∆ w = ζ ( xr ) in Ω and ∂ ν w = 0 on ∂ Ω, therefore (cid:90) Ω ζ ( xr ) K ( ζ ( xr )) = (cid:90) Ω ( − ∆ w ) w = (cid:90) Ω |∇ w | = (cid:107)∇ w (cid:107) L (Ω) . (5.11)On the other hand, multiplying − ∆ w = ζ ( xr ) in Ω by ζ ( xr ) and using H¨older’s inequality, r N (cid:107) ζ (cid:107) L (Ω) = (cid:90) Ω r | ζ ( xr ) | = (cid:90) Ω ( − ∆ w ) ζ ( xr ) = (cid:90) Ω ∇ w ∇ ( ζ ( xr )) = r − (cid:90) Ω r ∇ w ∇ ζ ( xr ) ≤ r − (cid:107)∇ w (cid:107) L (Ω) ( (cid:90) Ω r |∇ ζ ( xr ) | ) = r − (cid:107)∇ w (cid:107) L (Ω) r N (cid:107)∇ ζ (cid:107) L (Ω) , that is, (cid:107)∇ w (cid:107) L (Ω) ≥ r N (cid:107) ζ (cid:107) L (cid:107)∇ ζ (cid:107) L , and therefore, by (5.11), (cid:90) Ω ζ ( xr ) K ( ζ ( xr )) = (cid:107)∇ w (cid:107) L (Ω) ≥ r N (cid:107) ζ (cid:107) L (Ω) (cid:107)∇ ζ (cid:107) L (Ω) . (5.12)Then, by (5.10), (5.12), and observing that γ satisfies γ (1 /α + 1 /β ) + 2 = γ (cf. (5.1)) we obtain φ ( ϕ r , ψ r ) = (cid:90) Ω r | ϕ r | α α + | ψ r | β β − ψ r Kϕ r dx = (cid:90) Ω r r γ t αβα + β | ζ ( xr ) | α α + r γ t αβα + β | ζ ( xr ) | β β − r γ ( α + β ) tζ ( xr ) K ( ζ ( xr )) dx ≤ r γ + N (cid:90) Ω t αβα + β | ζ ( y ) | α α + t αβα + β | ζ ( y ) | β β dy − r γ ( α + β ) r N t (cid:107) ζ (cid:107) L (Ω) (cid:107)∇ ζ (cid:107) L (Ω) = r γ + N c < . (5.13)29e claim that E := φ ( f + ϕ r , g + ψ r ) − φ ( f, g ) <
0. Indeed, by (5.9), (cid:90) Ω r | f + ϕ r | α − | f | α − | ϕ r | α dx ≤ (cid:90) Ω r | o ( r γα ) + r γα t βα + β ζ ( xr ) | α − r γ | t βα + β ζ ( xr ) | α + o ( r γ ) dx ≤ r γ (cid:90) Ω r | o (1) + t βα + β ζ ( xr ) | α − | t βα + β ζ ( xr ) | α + o (1) dx ≤ r γ + N (cid:90) Ω | o (1) + t βα + β ζ ( y ) | α − | t βα + β ζ ( y ) | α + o (1) dy = o ( r γ + N ) (5.14)and analogously, (cid:82) Ω r | g + ψ r | β − | g | β − | ψ r | β dx = o ( r γ + N ) as r →
0. Furthermore, by Proposi-tion 5.3, (cid:90) Ω r | ψ r Kf | = (cid:90) Ω r | ψ r v | ≤ (cid:90) Ω r r qγq +1 | t αα + β ζ ( xr ) | o ( r γq +1 )= o ( r γ + N ) (cid:90) Ω | t αα + β ζ | = o ( r γ + N ) (5.15)as r →
0, and analogously, (cid:82) Ω r | ϕ r Kg | = o ( r γ + N ) as r →
0. Therefore, by (2.6), (5.13), (5.14),(5.15), and the fact that ϕ r ≡ R N \ Ω r , E = (cid:90) Ω | f + ϕ r | α − | f | α α + | g + ψ r | β − | g | β β − ( g + ψ r ) K ( f + ϕ r ) + gKf dx = φ ( ϕ r , ψ r ) + (cid:90) Ω r | f + ϕ r | α − | f | α − | ϕ r | α α + | g + ψ r | β − | g | β − | ψ r | β β − ψ r Kf − ϕ r Kg dx = φ ( ϕ r , ψ r ) + o ( r γ + N ) = r γ + N c + o ( r γ + N ) < r > Proof of Theorem 5.1.
Let ( f, g ) ∈ X be a critical point of φ in X and ( u, v ) := ( K p g, K q f ).By [18, Lemma 7.7], Proposition 2.4, and (1.1), we have that | v | q = | ∆ u | = 0 a.e. in u − (0)and | u | p = | ∆ v | = 0 a.e. in v − (0), therefore u − (0) = v − (0) a.e. Now, assume that (1.19)holds and ( f, g ) ∈ X is a global minimizer of φ in X . Then, by Proposition 5.4, we have that | u − (0) ∩ v − (0) | = 0 and thus | u − (0) | = | v − (0) | = 0, since u − (0) = v − (0) a.e. Remark 5.5. (Superlinear case) Let ( u, v ) ∈ [ C (Ω)] be a solution of (1.1) with p, q > pq > | u − (0) ∩ v − (0) | >
0, then ( u, v ) has a zero of infinite order ; indeed, if x ∈ Ω is a point ofdensity one for the set u − (0) ∩ v − (0) and γ >
0, then | u ( x ) | p +1 + | v ( x ) | q +1 = o ( | x − x | γ ) as x → x . A proof can be obtained by repeating almost word by word the proof of Proposition 5.3with the following changes: in Step 1, the functions u n , v n defined in (5.2) satisfy system (5.3)with (cid:101) A n := s n (sup | x | = s n | u ( x ) | p +1 + | v ( x ) | q +1 ) pq − p +1)( q +1) instead of A n as in (5.4), observe that (cid:101) A n → n → ∞ ; moreover, in Step 2, the functions (cid:101) u n , (cid:101) v n defined in (5.7) solve (instead of(5.8)) the system − ∆ (cid:101) u n = B n | (cid:101) v n | q − (cid:101) v n , − ∆ (cid:101) v n = B n | (cid:101) u n | p − (cid:101) u n , with B n = r γ pq − p +1)( q +1) n → n → ∞ . 30 .2 Simplicity of the zeros for radial solution Proof of Theorem 1.5.
Let t ( r ) = r − N if N ≥ t ( r ) = 1 − log( r ) if N = 2 ,b := lim s → δ t ( s ) ∈ (1 , ∞ ] ( b = ∞ if Ω = B ), J := [1 , b ), and ϕ, ψ : J → R satisfy ϕ ( t ( | x | )) = u ( x )and ψ ( t ( | x | )) = v ( x ). Then ( ϕ, ψ ) is a bounded solution of − ϕ (cid:48)(cid:48) = ζ | ψ | q − ψ, − ψ (cid:48)(cid:48) = ζ | ϕ | p − ϕ in J with ϕ (cid:48) = ψ (cid:48) = 0 on ∂J, (5.16)where ζ ( t ) := ( N − − t − N − N − ≥ N ≥ ζ ( t ) = e − t − ≥ N = 2.To prove (1.18) it suffices to show that | ϕ | + | ϕ (cid:48) | > | ψ | + | ψ (cid:48) | > J. (5.17)We argue by contradiction: assume there is x ∈ J such that ψ ( x ) = ψ (cid:48) ( x ) = 0 (5.18)Then, by (5.16), there is x ∈ ( x, b ) such that ϕ ( x ) = 0 . (5.19)Indeed, if | ϕ | > x, b ) then | ψ (cid:48)(cid:48) | > x, b ) and, in virtue of (5.18), ψ must be unboundedif b = ∞ or | ψ (cid:48) ( b ) | > b < ∞ , which is impossible by assumption, and therefore (5.19) follows.Now, it suffices to rule out the next three cases: a ) ϕ ( x ) = 0 , b ) ϕ ( x ) ϕ (cid:48) ( x ) ≥ , c ) ϕ ( x ) ϕ (cid:48) ( x ) < . Case a)
Assume that ϕ ( x ) = 0. Take the first integral Φ : J → R given byΦ := ϕ (cid:48) ψ (cid:48) + ζ ( | ϕ | p +1 p + 1 + | ψ | q +1 q + 1 ) , which satisfies Φ (cid:48) = ζ (cid:48) ( | ϕ | p +1 p + 1 + | ψ | q +1 q + 1 ) ≤ J, (5.20)by (5.16) and the fact that ζ is decreasing. Moreover, since ϕ ( x ) = 0, (5.18) implies that Φ( x ) = 0and therefore ϕ (cid:48) ψ (cid:48) ≤ − ζ ( | ϕ | p +1 p + 1 + | ψ | q +1 q + 1 ) ≤ x, b ) , (5.21)by (5.20). By (5.19), ϕ ( x ) = 0 for some x > x . We claim that ϕ ≡ x, x ]. By (5.16), thisimplies that also ψ ≡ x, x ], which contradicts the assumption that ϕ − (0) ∩ ψ − (0) hasempty interior, ruling out case a).To prove the claim, suppose that ϕ (cid:54)≡
0, and take y ∗ ∈ ( x, x ) such that | ϕ ( y ∗ ) | (cid:54) = 0.Without loss of generality, assume that ϕ ( y ∗ ) > ϕ (cid:48) ( y ∗ ) ψ (cid:48) ( y ∗ ) < y ∗ if necessary) we may assume that ϕ (cid:48) ( y ∗ ) >
0. But then, using (5.21) and a simple continuity argument, we have ϕ, ϕ (cid:48) > y ∗ , b ), in contradiction with (5.19). 31 ase b) Let ϕ ( x ) > ϕ (cid:48) ( x ) ≥ ϕ ( x ) < ϕ (cid:48) ( x ) ≤ x from (5.19) if necessary, we may assume that ϕ > x, x ), and by system (5.16)also − ψ (cid:48)(cid:48) > x, x ). By (5.18), this implies that ψ < x, x ) and therefore − ϕ (cid:48)(cid:48) < x, x ), i.e. , ϕ is convex in ( x, x ), which would contradict ϕ ( x ) = 0. Case c)
Finally, we rule out ϕ ( x ) ϕ (cid:48) ( x ) <
0. Suppose that ϕ ( x ) > ϕ (cid:48) ( x ) < ϕ ( x ) < ϕ (cid:48) ( x ) > ψ (cid:48)(cid:48) ( x ) <
0, which,combined with (5.18), yields ψ < w , x ) ⊆ (1 , x ). We claim that actually ψ < , x ), which yields a contradiction since this would imply ϕ (cid:48)(cid:48) > , x ) and thus ϕ (cid:48) (1) <
0, contradicting the boundary conditions in (5.16) .To prove the claim, assume w > ψ ( w ) = 0. Then, by the mean value theorem, thereis w ∈ ( w , x ) such that ψ (cid:48) ( w ) = 0. Since ψ (cid:48) ( x ) = 0 we may use again the mean value theoremto obtain w ∈ ( w , x ) such that ψ (cid:48)(cid:48) ( w ) = 0. But then ϕ ( w ) = 0 by (5.16), which is impossiblesince ϕ ( x ) < ϕ (cid:48) ( x ) >
0, and − ϕ (cid:48)(cid:48) = ζ | ψ | q − ψ > w , x ).Since we have reached a contradiction in all cases, we conclude that (5.18) cannot happen.The case ϕ ( x ) = ϕ (cid:48) ( x ) = 0 is completely analogous to (5.18) and therefore the claim (5.17)follows. Acknowledgments.
A. Salda˜na is supported by the Alexander von Humboldt Foundation,Germany. H. Tavares is partially supported by ERC Advanced Grant 2013 n. 339958 “ComplexPatterns for Strongly Interacting Dynamical Systems - COMPAT”. Both authors were partiallysupported by FCT/Portugal through UID/MAT/04459/2013. The graphs in Figure 1 and thenumerical approximations in Remark 3.6 were done using Mathematica 11.1, Wolfram ResearchInc. The authors would like to thank Tobias Weth for fruitful discussions and the anonymousreferee for the careful reading of the paper and helpful comments and suggestions.
References [1] A. Aftalion and F. Pacella. Qualitative properties of nodal solutions of semilinear ellipticequations in radially symmetric domains.
C. R. Math. Acad. Sci. Paris , 339(5):339–344,2004.[2] S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions ofelliptic partial differential equations satisfying general boundary conditions. I.
Comm. PureAppl. Math. , 12:623–727, 1959.[3] C.O. Alves and S. H. M. Soares. Singularly perturbed elliptic systems.
Nonlinear Anal. ,64(1):109–129, 2006.[4] A.I. ´Avila and J. Yang. On the existence and shape of least energy solutions for some ellipticsystems.
J. Differential Equations , 191(2):348–376, 2003.[5] T. Bartsch, T. Weth, and M. Willem. Partial symmetry of least energy nodal solutions tosome variational problems.
J. Anal. Math. , 96:1–18, 2005.[6] D. Bonheure, E. Moreira dos Santos, and H. Tavares. Hamiltonian elliptic systems: a guideto variational frameworks.
Port. Math. , 71(3-4):301–395, 2014.[7] D. Bonheure, E. Moreira dos Santos, and M. Ramos. Symmetry and symmetry breaking forground state solutions of some strongly coupled elliptic systems.
J. Funct. Anal. , 264(1):62–96, 2013. 328] D. Bonheure, E. Moreira dos Santos, M. Ramos, and H. Tavares. Existence and symmetryof least energy nodal solutions for Hamiltonian elliptic systems.
J. Math. Pures Appl. (9) ,104(6):1075–1107, 2015.[9] D. Bonheure, E. Serra, and P. Tilli. Radial positive solutions of elliptic systems with Neu-mann boundary conditions.
J. Funct. Anal. , 265(3):375–398, 2013.[10] F. Brock. Symmetry and monotonicity of solutions to some variational problems in cylindersand annuli.
Electron. J. Differential Equations , pages No. 108, 20 pp. (electronic), 2003.[11] J.A. Cardoso, J.M. do ´O, and E. Medeiros. Hamiltonian elliptic systems with nonlinearitiesof arbitrary growth.
Topol. Methods Nonlinear Anal. , 47(2):593–612, 2016.[12] D. Cassani and J. Zhang.
A priori estimates and positivity for semiclassical ground statesfor systems of critical Schr¨odinger equations in dimension two.
Comm. Partial DifferentialEquations , 42(5):655–702, 2017.[13] Ph. Cl´ement, D. G. de Figueiredo, and E. Mitidieri. Positive solutions of semilinear ellipticsystems.
Comm. Partial Differential Equations , 17(5-6):923–940, 1992.[14] Ph. Cl´ement and R. C. A. M. Van der Vorst. On a semilinear elliptic system.
DifferentialIntegral Equations , 8(6):1317–1329, 1995.[15] D.G. de Figueiredo. Semilinear elliptic systems: existence, multiplicity, symmetry of solu-tions. In
Handbook of differential equations: stationary partial differential equations. Vol.V , Handb. Differ. Equ., pages 1–48. Elsevier/North-Holland, Amsterdam, 2008.[16] D.G. de Figueiredo and J.-P. Gossez. Strict monotonicity of eigenvalues and unique contin-uation.
Comm. Partial Differential Equations , 17(1-2):339–346, 1992.[17] L.C. Evans and R.F. Gariepy.
Measure theory and fine properties of functions . Studies inAdvanced Mathematics. CRC Press, Boca Raton, FL, 1992.[18] D. Gilbarg and N.S. Trudinger.
Elliptic partial differential equations of second order . Classicsin Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.[19] P. Gir˜ao and T. Weth. The shape of extremal functions for Poincar´e-Sobolev-type inequal-ities in a ball.
J. Funct. Anal. , 237(1):194–223, 2006.[20] H. E. Gollwitzer. Nonoscillation theorems for a nonlinear differential equation.
Proc. Amer.Math. Soc. , 26:78–84, 1970.[21] J.-P. Gossez and J.-P. Loulit. A note on two notions of unique continuation,
Bull. Soc.Math. Belg. S´er. B , 45(3):257–268, 1993.[22] D. Jerison and C.E. Kenig. Unique continuation and absence of positive eigenvalues forSchr¨odinger operators.
Ann. of Math. (2) , 121(3):463–494, 1985.[23] S. Kesavan.
Symmetrization and applications.
Hackensack, NJ: World Scientific, 2006.[24] J. Lange Ferreira Melo and E. Moreira dos Santos. Critical and noncritical regions on thecritical hyperbola. In
Contributions to nonlinear elliptic equations and systems , volume 86 of
Progr. Nonlinear Differential Equations Appl. , pages 345–370. Birkh¨auser/Springer, Cham,2015. 3325] E. Parini and T. Weth. Existence, unique continuation and symmetry of least energy nodalsolutions to sublinear Neumann problems.
Math. Z. , 280(3-4):707–732, 2015.[26] L. A. Peletier and R. C. A. M. Van der Vorst. Existence and nonexistence of positivesolutions of nonlinear elliptic systems and the biharmonic equation.
Differential IntegralEquations , 5(4):747–767, 1992.[27] A. Pistoia and M. Ramos. Locating the peaks of the least energy solutions to an ellipticsystem with Neumann boundary conditions.
J. Differential Equations , 201(1):160–176, 2004.[28] M. Ramos and H. Tavares. Solutions with multiple spike patterns for an elliptic system.
Calc. Var. Partial Differential Equations , 31(1):1–25, 2008.[29] M. Ramos and J. Yang. Spike-layered solutions for an elliptic system with Neumann bound-ary conditions.
Trans. Amer. Math. Soc. , 357(8):3265–3284 (electronic), 2005.[30] G.M. Rassias and T.M. Rassias, editors.
Differential geometry, calculus of variations, andtheir applications , volume 100 of
Lecture Notes in Pure and Applied Mathematics . MarcelDekker, Inc., New York, 1985.[31] B. Ruf. Superlinear elliptic equations and systems. In
Handbook of differential equa-tions: stationary partial differential equations. Vol. V , Handb. Differ. Equ., pages 211–276.Elsevier/North-Holland, Amsterdam, 2008.[32] A. Salda˜na and T. Weth. Asymptotic axial symmetry of solutions of parabolic equations inbounded radial domains.
J. Evol. Equ. , 12(3):697–712, 2012.[33] B. Sirakov. On the existence of solutions of Hamiltonian elliptic systems in R N . Adv.Differential Equations , 5(10-12):1445–1464, 2000.[34] D. Smets and M. Willem. Partial symmetry and asymptotic behavior for some ellipticvariational problems.
Calc. Var. Partial Differential Equations , 18(1):57–75, 2003.[35] T. Weth. Symmetry of solutions to variational problems for nonlinear elliptic equations viareflection methods.
Jahresber. Dtsch. Math.-Ver. , 112(3):119–158, 2010.[36] J. S. W. Wong. On the generalized Emden-Fowler equation.
SIAM Rev. , 17:339–360, 1975.[37] J. Zeng. The estimates on the energy functional of an elliptic system with Neumann bound-ary conditions.