A supercritical elliptic equation in the annulus
Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris, Tobias Weth
aa r X i v : . [ m a t h . A P ] F e b A SUPERCRITICAL ELLIPTIC EQUATION IN THE ANNULUS
ALBERTO BOSCAGGIN, FRANCESCA COLASUONNO, BENEDETTA NORIS,AND TOBIAS WETH
Abstract.
By a combination of variational and topological techniques in thepresence of invariant cones, we detect a new type of positive axially symmetricsolutions of the Dirichlet problem for the elliptic equation − ∆ u + u = a ( x ) | u | p − u in an annulus A ⊂ R N ( N ≥ p > a ( x ) is an axially symmetric but possibly nonradial function with additionalsymmetry and monotonicity properties, which are shared by the solution u weconstruct. In the case where a equals a positive constant, we obtain nonradialsolutions in the case where the exponent p is large or when the annulus A islarge with fixed width. Introduction
In the present paper, we are concerned with the nonlinear elliptic equation − ∆ u + u = a ( x ) | u | p − u (1.1)in the case where N ≥
3, the nonlinearity is (possibly) supercritical, i.e., p > ∗ := 2 N/ ( N − x a ( x ) is a positive weight function. Due to the lack ofembeddings of the Sobolev space H ( R N ) into the space L p ( R N ) in this case, theequation (1.1) does not admit a variational framework in H ( R N ). The same is truefor the Dirichlet and Neumann problem for (1.1) in a bounded domain Ω ⊂ R N ,as neither H (Ω) nor H (Ω) is embedded in L p (Ω) if p > ∗ . Hence standardvariational methods do not apply in these cases. Incidentally, let us recall that,due to Pohozaev identity, the Dirichlet problem for (1.1) does not admit nontrivialsolutions in a bounded star-shaped domain Ω if p ≥ ∗ and a is positive constantweight function, see e.g. [28]. On the other hand, while no obstruction for thesolvability of the Dirichlet problem for (1.1) is known in the case of topologicallynontrivial domains, no general approach is available to study this problem in the fullsupercritical regime. Among others, relevant existence results for specific relatedcritical and slightly supercritical Dirichlet problems can be found in [1, 18, 24, 25].In the present paper, we wish to show that the combination of variational andtopological methods in the spaces H (Ω) and C (Ω) can yield existence of positivesolutions of the Dirichlet problem for (1.1) in the case where a cone of functionswith suitable invariance properties can be found. The presence of invariant cones,characterized by monotonicity properties of functions, has already been used in [3,12,26] to construct solutions of the Neumann problem for the supercritical equation(1.1) in specific domains, see also [4,5,11] for related results. More precisely, in [3–5, Mathematics Subject Classification.
Key words and phrases.
Supercritical elliptic equations, Variational and topological methods,Invariant cones, High Morse index solutions, Axially symmetric solutions. ⊂ R N and a radial and a radially increasing function a isconsidered, while [12] is devoted to domains given as a product of lower-dimensionalballs and a function a with associated symmetry and monotonicity properties. Akey difference between these papers dealing with the Neumann problem and ourpresent work is the fact that, by construction, the solutions found in [3–5,11,12,26]attain their maximum on the boundary of the underlying domain, which cannotbe realized for the corresponding Dirichlet problem. We also refer to [16, 17] forfurther existence results for supercritical Neumann problems.In the present paper, we focus on the problem − ∆ u + u = a ( x ) | u | p − u in A,u >
A,u = 0 on ∂A, (1.2)where A is a bounded N -dimensional annulus A := { x ∈ R N : R < | x | < R } with N ≥ < R < R < ∞ ). As it is well known, the existence of a radialsolution to (1.2) can be easily proved, for any p >
2, assuming a is a radial positivebounded function. In this paper, we propose instead to investigate problem (1.2)for a certain class of possibly nonradial but axially symmetric weight functions a .We point out that the restriction to axially symmetric functions alone does nothelp to overcome the lack of a variational structure and compactness properties inthe supercritical case, since axially symmetric functions may concentrate on thesymmetry axis which has a nonempty intersection with the annulus A .We now state our result precisely and introduce some notation. Assuming with-out loss of generality that the axis of symmetry is the x N -axis, we call a functionon A axially symmetric if it only depends on r = | x | ∈ [ R , R ] and θ = arcsin (cid:16) x N r (cid:17) ∈ h − π , π i . Hence every axially symmetric function u on A can be written as u ( x ) = u ( | x | , arcsin ( x N /r )) with a function u : [ R , R ] × h − π , π i → R .To describe further related symmetry and monotonocity properties, we introducethe cone b K := u ∈ C ( A ) : u = u ( r, θ ) , u ≥ A, u ( r, θ ) = u ( r, − θ ) in [ R , R ] × (0 , π/ , u θ ( r, θ ) ≤ R , R ] × (0 , π/ . , (1.3)where u θ stands for the partial derivative with respect to the variable θ . Noticethat a function u ∈ b K satisfies also u θ ( r, θ ) ≥ R , R ] × ( − π/ , . We also set K := { u ∈ b K : u (cid:12)(cid:12) ∂A ≡ } . (1.4)Hence K is the intersection of b K with the function space C ( A ) := { u ∈ C ( A ) : u (cid:12)(cid:12) ∂A ≡ } ⊂ H ( A ) , (1.5)which will play a central role in the variational approach we propose in this paper.With this notation, we assume that a ∈ b K , a > A, (1.6) and we will show that (1.2) admits a nontrivial solution belonging to K and enjoin-ing a suitable minimality property. To describe it precisely, we define the functional I : C ( A ) → R , I ( u ) := 12 Z A ( |∇ u | + u ) dx − p Z A a ( x ) | u | p dx, (1.7)which is well known to be well-defined and of class C , since p >
2. Within thecone K , we also consider the following Nehari-type set N K := { u ∈ K : u , I ′ ( u ) u = 0 } and the Nehari value c I := inf u ∈N K I ( u ) . (1.8)With this notation, our first main result now reads as follows. Theorem 1.1.
Let N ≥ , p > , and suppose that the function a satisfies (1.6).Then we have c I > , and c I is attained in N K . Moreover, every minimizer u ∈ N K of I (cid:12)(cid:12) N K is a nontrivial solution of (1.2) belonging to K . Hereafter we will call every minimizer u ∈ N K of I (cid:12)(cid:12) N K a K -ground state solutionof (1.2) .In the case where a is a non-radial function, every solution of (1.2) is nonradial,and Theorem 1.1 yields a new existence result for such solutions in the case ofcritical or supercritical exponents p > a is radial andradially increasing allows to overcome the lack of a global variational structure andof compactness by restricting variational arguments to the cone of positive radialand radially increasing functions. In a similar spirit, we will exploit here the factthat a ∈ b K to gain some compactness within the cone K and then prove, by theuse of a combination of topological and variational techniques, the existence of asolution u to (1.2) in K .To explain the argument in more detail, we first notice that the minimum value c I can be reinterpreted as a minimax value of mountain pass-type, cf. Lemma 2.5.Hence, we develop a mountain pass-type argument for the functional I in the cone K , by replacing the usual gradient flow with ddt η ( t, u ) = − (Id − T )( η ( t, u )) , where T : C ( A ) → C ( A ) is the operator defined by T ( u ) := ( − ∆ + Id) − ( a ( x ) | u | p − u ) . Notice that, with this approach, solutions will be provided as fixed points of theoperator T . A major difficulty lies in the fact that the functional I is of class C in the space C ( A ), whereas the compactness properties are available with respectto the H ( A )-norm (see the Palais-Smale type condition proved in Lemma 3.1).In order to overcome this obstacle, we adopt a dynamical system point of view,partially inspired from [2]. More precisely, via the descent flow, we manage toconstruct a sequence belonging to the boundary of a certain domain of attraction,that converges to a fixed point of T , that is a solution of the problem. A. BOSCAGGIN, F. COLASUONNO, B. NORIS, AND T. WETH
In the following, we wish to discuss the important special case where a is aconstant function. In this case we may, by renormalization, assume that a ≡ − ∆ u + u = | u | p − u in A,u >
A,u = 0 on ∂A (1.9)which has received widespread attention with regard to the existence and shape ofradial and nonradial solutions. As already observed, a radial solution exists for any p > A is an annulus with a small hole and the nonlinearity is subcritical, see [21]. Onthe other hand, for expanding annuli with fixed difference of radii, the existence ofmultiple nonradial solutions has been proved in [10] in the two-dimensional case, andin [7,8] for N ≥ K -ground state solutions of (1.9), as detected by Theorem 1.1, arein fact nonradial, and thus we obtain new existence results for nonradial solutionsin the case of critical and supercritical exponents.The first result is devoted to large exponents p . Theorem 1.2.
Let N ≥ . Then there exists a value p ∗ > with the property thatevery K -ground state solution of (1.9) is nonradial for p > p ∗ .In particular, (1.9) admits a positive nonradial but axially symmetric solution for p > p ∗ . Theorem 1.2 complements a result in [20] on local bifurcation of nonradial solu-tions for problem (1.9). More precisely, it is shown in [20, Theorem 1.7] that thereexists an ordered sequence { p k } k with lim k →∞ p k = ∞ of bifurcation points in thesense that, for every fixed k ∈ N , there exists a sequence { u k,ℓ } ℓ of nonradial solu-tions of (1.9) for a corresponding sequence of exponents { p k,ℓ } ℓ with lim ℓ →∞ p k,ℓ = p k and with the property that u k,ℓ converges to the unique radial positive solution of(1.9) with p = p k as ℓ → ∞ . Theorem 1.2 suggests that one of these bifurcationpoints corresponds to a global branch covering the unbounded interval ( p ∗ , ∞ ) ofexponents p .Our next and final main result holds for arbitrary exponents p > A R := { x ∈ R N : R < | x | < R + 1 } for R > Theorem 1.3.
Let N ≥ and p > . Then there exists a value R ∗ > , dependingon p and the dimension N , with the property that every K -ground state solution of(1.9) in A = A R is nonradial for R > R ∗ . In particular, (1.9) admits a positive nonradial but axially symmetric solution in A = A R for R > R ∗ . We note that, in the same way as Theorem 1.2 complements [20, Theorem 1.7],Theorem 1.3 also complements similar bifurcation results given in [20, Theorems 1.3and 1.4] with respect to the parameter R in the problem (1.9) on A = A R .We briefly comment on the proof of Theorems 1.2 and 1.3. As a consequenceof Theorem 1.1, it suffices to show that the unique radial solution of (1.9) cannotbe a K -ground state solution under the given assumptions. For this, we will showa specific instability property of the unique radial solution of (1.9) with respect tothe cone K , see Proposition 4.2 below. We use some results from [20] in the proofof this instability property.We also remark that the solutions obtained both in Theorem 1.2 and in The-orem 1.3 have Morse index greater than N . This follows from [23, Theorem 1.1]and the fact that nonradial functions in K are axially symmetric but not foliatedSchwarz symmetric.The paper is organized as follows. In Section 2, we collect some preliminaryresults and a priori estimates in the cone K , see in particular Lemma 2.3: it isinteresting to observe that its proof uses trace inequalities and embeddings theoremsfor fractional Sobolev spaces. In Section 3, we prove the main result of the paper,Theorem 1.1. Finally, in Section 4, we deal with the case a ≡ const . and proveTheorems 1.2 and 1.3. 2. Preliminary results
The linear problem in the cone.
Let us consider the set K defined in (1.4).It is easy to verify that it is a closed convex cone in C ( A ), that is:(i) if u ∈ K and λ > λu ∈ K ;(ii) if u, v ∈ K then u + v ∈ K ;(iii) if u, − u ∈ K then u ≡ K is closed in the C -topology.We begin with the following auxiliary result; it deals with the linear problemwith right-hand side belonging to K . Lemma 2.1.
For any h ∈ K , the unique solution u to the linear problem ( − ∆ u + u = h in A,u = 0 on ∂A (2.1) belongs to K .Proof. Let us first notice that, since h ∈ C ( A ) ⊂ C ,α ( A ), by elliptic regularitywe have u ∈ C ,α ( A ) and thus u ∈ C ( A ), as well. By uniqueness and thanks tothe fact that the problem (i.e., the operator, the right-hand side, and the domain)is invariant under the action of the group O ( N − × O (1), the solution u is suchthat u = u ( r, θ ) and u ( r, θ ) = u ( r, − θ ) for θ ∈ ( − π/ , π/ h ≥
0, also u ≥ A .In order to prove the monotonicity with respect to θ , we recall the expression ofthe Laplacian of an axially symmetric function:∆ u = u rr + N − r u r + 1 r ∆ S N − u , (2.2) A. BOSCAGGIN, F. COLASUONNO, B. NORIS, AND T. WETH where ∆ S N − is the Laplace-Beltrami operator on the ( N − S N − u = 1cos N − θ ∂ θ (cid:0) cos N − θ u θ (cid:1) . Therefore, if we perform a partial derivative in θ for the equation in (2.1), we getthe pointwise equation − u θrr − N − r u θr + N − r θ u θ + N − r tan θ u θθ + 1 r u θθ + u θ = h θ , for ( r, θ ) ∈ ( R , R ) × ( − π/ , π/ u θ ( x ) , h θ ( x ) by the relations u θ ( x ) = u θ (cid:16) | x | , arcsin (cid:16) x N r (cid:17)(cid:17) , h θ ( x ) = h θ (cid:16) | x | , arcsin (cid:16) x N r (cid:17)(cid:17) , and noticing that r cos θ = x + . . . + x N − , we can rewrite the previous expressionas − ∆ u θ + (cid:16) N − x + . . . + x N − + 1 (cid:17) u θ = h θ in ˜ A := { x ∈ A : x + . . . + x N − = 0 } . We now wish to show that u θ ≤ A + := ˜ A ∩ { x N > } . (2.3)For this we first note that u θ = 0 on ∂ ˜ A + . Indeed, being u regular and axiallysymmetric with respect to the x N -axis, we deduce that u θ ( r, π/
2) = 0 for every r ∈ [ R , R ], so that u θ vanishes along the x N -axis. Moreover, since u = 0 on ∂A , and u θ ( r, θ ) is the derivative of u in the tangential direction to ∂A , we have u θ ( r, θ ) = 0on { R , R } × (0 , π/ u θ = 0 on ∂A . Finally, since u is axially symmetricand even with respect to θ , again by regularity we have u θ ( r, + ) = − u θ ( r, − ) = u θ ( r,
0) for every r ∈ [ R , R ], and so u θ ( r,
0) = 0 for every r ∈ [ R , R ]. Hence u θ = 0 on ∂ ( A ∩ { x N > } ).In sum, u θ satisfies the problem − ∆ u θ + (cid:16) N − x + . . . + x N − + 1 (cid:17) u θ = h θ ≤ A + u θ = 0 on ∂ ˜ A + . (2.4)Due to the singularity of the equation on the x N -axis, we cannot apply the weakmaximum principle directly to deduce that u θ ≤
0. Instead, we let ε > v := ( u θ − ε ) + on ˜ A + . By (2.4) and since u θ ∈ C ( ˜ A + ), thefunction v ∈ H ( ˜ A + ) has compact support in ˜ A + . Hence we may multiply (2.4)with v and integrate by parts, obtaining the inequality Z ˜ A + (cid:16) |∇ v | + (cid:16) N − x + . . . + x N − + 1 (cid:17) v (cid:17) dx ≤ Z ˜ A + (cid:16) ∇ u θ · ∇ v + (cid:16) N − x + . . . + x N − + 1 (cid:17) u θ v (cid:17) dx = Z ˜ A + (cid:16) − ∆ u θ + (cid:16) N − x + . . . + x N − + 1 (cid:17) u θ (cid:17) v dx = Z ˜ A + h θ v dx ≤ . (2.5)Here, the integration by parts in the second step is justified by approximating v in the H -norm by a sequence of functions ( v n ) n ⊂ C ∞ c (Ω), where Ω ⊂ ˜ A + is acompactly contained subdomain containing the support of v . Now (2.5) implies that v = ( u θ − ε ) + ≡ A + . Since ε > u is even with respect to θ , then u θ ≥ θ ∈ ( − π/ ,
0) and theproof is concluded. (cid:3)
A priori estimates in the cone.
In this section, we deal with the largercone e K := u ∈ H ( A ) : u = u ( r, θ ) , u ≥ A, u ( r, θ ) = u ( r, − θ ) a.e. in ( R , R ) × (0 , π/ , u θ ( r, θ ) ≤ R , R ) × (0 , π/ . . where u θ denotes the weak derivative of u with respect to θ . Notice that, be-ing u ∈ H ( A ), we have that u ∈ H (( R , R ) × ( − π/ , π/ u θ ∈ L (( R , R ) × ( − π/ , π/ u θ appearing in (1.3) makes sense.Of course, K = e K ∩ C ( A ). The fact that e K is a cone is easily verified (cf (i)-(iii)at the beginning of the previous section). Below, we explicitly prove that e K isclosed with respect to the H -topology. Lemma 2.2. e K is closed with respect to the H ( A ) -norm; as a consequence, it isweakly closed.Proof. Let { u n } n ⊂ e K and u ∈ H ( A ) be such that u n → u in H ( A ) as n → ∞ .Clearly, u is axially symmetric, non-negative and even with respect to θ by pointwisealmost everywhere convergence up to a subsequence. Let us check that u θ ≤ ≥ ∂ u n ∂θ = ∇ u n · ∂x∂θ → ∇ u · ∂x∂θ = u θ almost everywhere, as n → ∞ . Then, u ∈ e K , proving that e K is closed in the strong H -topology. Since a cone is a convex set, we conclude that e K is weakly closed, aswell. (cid:3) In the following we denote D := A ∩ { x N = 0 } and we use the notation x =( x ′ , x N ) for every x ∈ R N so that x = ( x ′ ,
0) for every x ∈ D . The main result ofthis section is the following a-priori bound for functions in the cone e K . Lemma 2.3. e K ⊂ L q ( A ) for every q ≥ . Moreover, for every q ≥ there exists apositive constant C ( q ) such that k u k L q ( A ) ≤ C ( q ) k u k H ( A ) for every u ∈ e K . (2.6) Proof.
Let ˜ u be the trivial extension to zero of u outside A . On the one hand, bythe trace inequality, we have[˜ u ( x ′ , H / ( R N − ) ≤ C k ˜ u k H ( R N + ) ≤ C k ˜ u k H ( R N ) = C k u k H ( A ) . (2.7)On the other hand, by [14, Lemma 5.1] with n = N − s = 1 /
2, since ˜ u ( x ′ ,
0) is aradial function in R N − , we obtain for every c > − (cid:18)Z R N − | x ′ | c | ˜ u ( x ′ , | ∗ c dx ′ (cid:19) / ∗ c ≤ C [˜ u ( x ′ , H / ( R N − ) , (2.8) A. BOSCAGGIN, F. COLASUONNO, B. NORIS, AND T. WETH where 2 ∗ c := N − c ) N − . Notice that [14, Lemma 5.1] is stated for C ∞ c ( R N − ) radialfunctions, but it can be extended, by a density argument, to H / ( R N − ) radialfunctions. Moreover, being R > Z R N − | x ′ | c | ˜ u ( x ′ , | ∗ c dx ′ = Z D | x ′ | c | u ( x ′ , | ∗ c dx ′ ≥ R c k u ( · , k ∗ c L ∗ c ( D ) . (2.9)Since 2 ∗ c → ∞ as c → ∞ , combining (2.7), (2.8) and (2.9), we get the existence ofa constant C ( q ) such that k u ( · , k L q ( D ) ≤ C ( q ) k u k H ( A ) for every u ∈ e K . (2.10)By the axial symmetry and the monotonicity properties of u ∈ e K , we deduce Z A | u | q dx = Z R R Z π − π | u ( r, θ ) | q r N − | cos θ | dr dθ ≤ Z R R Z π − π | u ( r, | q R r N − dr dθ = R π Z R R | u ( r, | q r N − dr = R π Z D | u ( · , | q dx ′ . Combining the last inequality with (2.10) we have the desired estimate. (cid:3)
The fixed point operator T . Hereafter, let p > T : e K ∪ C ( A ) → H ( A ) T ( u ) := ( − ∆ + Id) − ( a ( x ) | u | p − u ) , namely T ( u ) = v is the unique H ( A ) function satisfying Z A ( ∇ v · ∇ ϕ + vϕ ) dx = Z A a ( x ) | u | p − uϕ dx for every ϕ ∈ C ∞ c ( A ) . (2.11)This definition is clearly well posed when u ∈ C ( A ) since a ( x ) | u | p − u ∈ C ( A ).On the other hand, when u ∈ e K , by Lemma 2.3 we have u p − ∈ L q ( A ) for every q ≥ . This, in particular, implies that a ( x ) | u | p − u = a ( x ) u p − ∈ L ( A ) . (2.12)so that T ( u ) is well-defined, again, with T ( u ) ∈ H ( A ).We also observe that T ( K ) ⊂ K . (2.13)Indeed, thanks to (1.6), we have that a ( x ) u p − ∈ K for every u ∈ K and so, byLemma 2.1, T ( u ) ∈ K .We now prove that T , when restricted to K , has suitable continuity and com-pactness properties. Proposition 2.4.
Let { u n } n ⊂ K be such that u n ⇀ u weakly in H ( A ) for some u ∈ e K . Then T ( u ) ∈ K and T ( u n ) → T ( u ) in C ( A ) . Proof.
Let us first prove that T ( u n ) → T ( u ) in H ( A ). By the definition (2.11) of T , we have k T ( u n ) − T ( u ) k H ( A ) = Z A a ( x )( u p − n − u p − )( T ( u n ) − T ( u )) dx ≤ ( p − k a k L ∞ ( A ) Z A ( u n + u ) p − | u n − u || T ( u n ) − T ( u ) | dx, where we used the inequality (see [13]) (cid:12)(cid:12) ξ p − − η p − (cid:12)(cid:12) ≤ ( p −
1) ( ξ + η ) p − | ξ − η | for every ξ, η ∈ R + , p ≥ . Let α > β > max n N, p − o be such that1 α + 1 β + 12 = 1 . By the choice of β , it results α < ∗ and β ( p − >
1, and so, Lemma 2.3 and theH¨older inequality applied to the previous expression provide k T ( u n ) − T ( u ) k H ( A ) ≤ C (cid:0) k u n k L β ( p − ( A ) + k u k L β ( p − ( A ) (cid:1) p − k u n − u k L α ( A ) k T ( u n ) − T ( u ) k H ( A ) , with C = ( p − k a k L ∞ ( A ) . Since { u n } n is weakly convergent, it is bounded in the H ( A )-norm. Hence, by combining relation (2.6) with the previous estimate wededuce k T ( u n ) − T ( u ) k H ( A ) ≤ C ′ k u n − u k L α ( A ) , for a constant C ′ not depending on n . Hence, the weak convergence u n ⇀ u in H ( A ) and α < ∗ imply lim n → + ∞ k T ( u n ) − T ( u ) k H ( A ) = 0 , as desired. Since e K is closed, T ( u ) ∈ e K and, by elliptic regularity, T ( u ) ∈ K .Now, let v n := T ( u n ) for n ∈ N and v := T ( u ). By the first part of the proof,we know that v n → v in H ( A ). Hence it suffices to show that the sequence { v n } n is relatively compact in C ( A ). Let q ∈ (1 , ∞ ). By (1.6), Lemma 2.3, and theboundedness of { u n } n in H ( A ), we see that k f n k L q ( A ) ≤ C q for n ∈ N with f n := a ( x ) u p − n (2.14)and some constant C q >
0. Since the functions v n ∈ H ( A ) solve − ∆ v n + v n = f n in A , elliptic regularity estimates show that k v n k W ,q ( A ) ≤ C ′ q for n ∈ N with some constant C ′ q >
0. Since, as ∂A is smooth, we have a compact embedding W ,q ( A ) ∩ H ( A ) ֒ → C ( A ) for q > N , we conclude that the sequence { v n } n isrelatively compact in C ( A ). (cid:3) An equivalent minimax characterization.
We define the minimax value d I := inf u ∈K u sup t> I ( tu ) , (2.15)where the functional I is defined by (1.7). By elementary properties of I , it is easyto see that for every function u ∈ C ( A ) \{ } there exists precisely one critical point t u > t I ( tu ) which is the global maximum of this function on(0 , ∞ ) (see for example [28, Chapter 4]). More precisely, t u = k u k H ( A ) R A a ( x ) | u | p dx ! / ( p − (2.16)and it is straightforward to verify that t u u ∈ N K . Lemma 2.5.
The value c I introduced in (1.8) coincides with d I .Proof. For every u ∈ K \ { } , t u u ∈ N K , with t u as in (2.16), thus implying that d I = inf u ∈K\{ } I ( t u u ) ≥ c I . In order to prove the opposite inequality, notice that the map H : u ∈ K ∩ S t u u ∈ N K , with S = { u ∈ H ( A ) : k u k H ( A ) = 1 } , is bijective. Indeed, the map v ∈ N K v/ k v k H ( A ) ∈ K ∩ S is the inverse of H by the uniqueness of t u and bythe fact that t u = 1 if and only if u ∈ N K . Therefore d I ≤ inf u ∈K∩S sup t> I ( tu ) = inf u ∈K∩S I ( H ( u )) = c I . (cid:3) Proof of the main result
In this section, we give the proof of Theorem 1.1 via a critical point theoryapproach in the space C ( A ) and more precisely in the cone K introduced in (1.4).Although K is a subset of C ( A ), we emphasize that our argument requires the useof both the C and the H topology. As already mentioned in the introduction, thesolution is found as a fixed point of the operator T given in (2.11), by means of adynamical system point of view applied to a suitable descent flow. For the reader’sconvenience, we divide the section in several subsections.3.1. Compactness and geometry of the functional I . Let us consider thefunctional I defined in (1.7). Recalling the definition of the operator T given in(2.11), we observe that I ′ ( u ) v = Z A ( ∇ u · ∇ v + uv − a ( x ) | u | p − uv ) dx = h u − T ( u ) , v i H ( A ) , for every u, v ∈ C ( A ). We first show that I satisfies a Palais-Smale type conditionin K , with respect to the H -norm. Lemma 3.1.
Let { u n } n ⊂ K be such that (i) { I ( u n ) } n is bounded; (ii) lim n → + ∞ k u n − T ( u n ) k H ( A ) = 0 .Then there exist a subsequence { u n k } k and u ∈ K such that lim k → + ∞ k u n k − u k H ( A ) = 0 and u = T ( u ) . Proof.
By assumption (i) there exists a constant
C > C ≥ I ( u n ) = (cid:18) − p (cid:19) k u n k H ( A ) + 1 p (cid:20) k u n k H ( A ) − Z A a ( x ) u pn dx (cid:21) = (cid:18) − p (cid:19) k u n k H ( A ) − p h T ( u n ) − u n , u n i H ( A ) ≥ (cid:18) − p (cid:19) k u n k H ( A ) − p k T ( u n ) − u n k H ( A ) k u n k H ( A ) , (3.1)for every n ≥
1, where we also used the definition of T (see (2.11)) and the Cauchy-Schwarz inequality. Now, the last inequality combined with assumption (ii) impliesthat the sequence { u n } is bounded in the H ( A )-norm. We deduce the existenceof a subsequence { u n k } k and u ∈ H ( A ) such that u n k ⇀ u weakly in H ( A ) as k → + ∞ . By Lemma 2.2, u ∈ e K . Then Proposition 2.4 provides T ( u ) ∈ K andlim k → + ∞ k T ( u n k ) − T ( u ) k H ( A ) = 0 . In turn, using again assumption (ii), we obtain o (1) = k T ( u n k ) − u n k k H ( A ) = k T ( u ) − u n k k H ( A ) + o (1)as k → + ∞ , from which we deduce both that u n k converges to u strongly in H ( A )and that u = T ( u ). In particular, u ∈ K . (cid:3) In the next lemma we prove that I has a mountain pass type geometry. Lemma 3.2.
There exists α > with the property that for B α ( K ) := { u ∈ K : k u k H ( A ) < α } , S α ( K ) := { u ∈ K : k u k H ( A ) = α } we have: (i) I is nonnegative on B α ( K ) . (ii) ρ α := inf u ∈ S α ( K ) I ( u ) > . Proof.
Let u ∈ K . By Lemma 2.3 with q = p , we get I ( u ) ≥ k u k H ( A ) − p k a k L ∞ ( A ) k u k pL p ( A ) ≥ k u k H ( A ) − C ( p ) p p k a k L ∞ ( A ) k u k pH ( A ) , whence (i) and (ii) follow immediately, being p > (cid:3) A descent flow in the cone.
In the following, we develop a descent flowargument inside the cone K . For every v ∈ C ( A ), letΦ( v ) := v − T ( v ) , then Φ : C ( A ) → C ( A ) is locally Lipschitz. For every u ∈ C ( A ), let η ( t, u ) bethe unique solution of the following initial value problem ( ddt η ( t, u ) = − Φ( η ( t, u )) η (0 , u ) = u, (3.2)defined on its maximal interval [0 , T max ( u )).We observe that T max ( u ) may be finite for some u , due to the fact that theright hand side of (3.2) is not normalised. We made this choice because a C normalisation, that would have ensured existence of η ( t, u ) for all times t for every u ∈ C ( A ), would have invalidated estimate (3.3) below. Remark . Being Φ locally Lipschitz, the solution of (3.2) depends continuouslyon the initial data (see for example [15]). That is, for every u ∈ C ( A ), for every¯ t < T max ( u ) and for every { v n } ⊂ C ( A ) such that k v n − u k C ( A ) →
0, there exists¯ n ≥ n ≥ ¯ n , the solution η ( t, v n ) is defined for every t ∈ [0 , ¯ t ]and sup t ∈ [0 , ¯ t ] k η ( t, v n ) − η ( t, u ) k C ( A ) → , as n → + ∞ . Let us show that the cone K is invariant under the action of the flow η . Lemma 3.4.
For every u ∈ K and for every t < T max ( u ) , η ( t, u ) ∈ K .Proof. The proof is analogous to the one in [3, Lemma 4.5] (see also [9, 11]). Webriefly sketch it below for the sake of completeness. For every n ∈ N , we considerthe approximation of the flow line t ∈ [0 , T max ( u )) η ( t, u ) given by the Eulerpolygonal t ∈ [0 , T max ( u )) η n ( t, u ). The vertices of such polygonal η n are definedby the following recurrence formula: ( η n (0 , u ) = 0 η n ( t i +1 , u ) := η n ( t i , u ) − T max ( u ) n Φ( η n ( t i , u )) for all i = 0 , . . . , n − , where t i := in T max ( u ) for every i = 0 , . . . , k . Recalling the definition of Φ, since T preserves the cone K , it is easy to prove that the vertices of the polygonal η n belong to K by convexity. Hence, again by convexity, η n ([0 , T max ( u )) , u ) ⊂ K forevery n . Finally, being Φ locally Lipschitz, the following convergence holds forevery t ∈ [0 , T max ( u )) lim n → + ∞ k η n ( t, u ) − η ( t, u ) k C ( A ) = 0 . The statement then follows immediately, being K closed in the C -topology. (cid:3) In the next lemma we prove that the energy functional I decreases along thetrajectories η ( · , u ). Moreover, we give a condition on u sufficient to guarantee theglobal existence of η ( · , u ) and to construct a related Palais-Smale sequence. Lemma 3.5.
Let u ∈ C ( A ) . Then we have ddt I ( η ( t, u )) = −k Φ( η ( t, u )) k H ( A ) for every t ∈ (0 , T max ( u )) . (3.3) Consequently, the functional I is nonincreasing along the trajectories of η . More-over, if u ∈ K and c u := lim t → T max ( u ) I ( η ( t, u )) > −∞ , (3.4) then T max ( u ) = ∞ , and there exists a sequence { s n } n ⊂ (0 , + ∞ ) such that lim n → + ∞ s n =+ ∞ and lim n → + ∞ k Φ( w n ) k H ( A ) = 0 (3.5) with w n := η ( s n , u ) for n ≥ . (3.6) Proof.
Let T ∗ := T max ( u ). For t ∈ (0 , T ∗ ) we have ddt I ( η ( t, u )) = Z A h(cid:2) ∇ η ( t, u ) · ∇ (cid:0) T ( η ( t, u )) − η ( t, u ) (cid:1) + η ( t, u ) (cid:0) T ( η ( t, u )) − η ( t, u ) (cid:1)(cid:3) − Z A a ( x ) | η ( t, u ) | p − η ( t, u ) (cid:0) T ( η ( t, u )) − η ( t, u ) (cid:1)i dx = h η ( t, u ) , T ( η ( t, u )) − η ( t, u ) i H ( A ) − h T ( η ( t, u )) , T ( η ( t, u )) − η ( t, u ) i H ( A ) = −k T ( η ( t, u )) − η ( t, u ) k H ( A ) , as claimed in (3.3).Next we assume that (3.4) holds. In order to prove that T ∗ = ∞ we proceed bycontradiction, thus assuming T ∗ < ∞ , and consequentlylim t → T −∗ k η ( t, u ) k C ( A ) = + ∞ . (3.7)For 0 ≤ s < t < T ∗ , we then have, using (3.3), k η ( t, u ) − η ( s, u ) k H ( A ) ≤ Z ts (cid:13)(cid:13)(cid:13)(cid:13) ddτ η ( τ, u ) (cid:13)(cid:13)(cid:13)(cid:13) H ( A ) dτ = Z ts r − ddτ I ( η ( τ, u )) dτ ≤ √ t − s s − Z ts ddτ I ( η ( τ, u )) dτ = √ t − s [ I ( η ( s, u )) − I ( η ( t, u ))] ≤ √ t − s [ I ( u ) − c u ] . Since, by our contradiction assumption, T ∗ < ∞ , we deduce that for every sequence { t n } n ⊂ (0 , T ∗ ) such that t n → T −∗ as n → ∞ , { η ( t n , u ) } n is a Cauchy sequence.This implies that there exists w ∈ H ( A ) such thatlim t → T ∗ k η ( t, u ) − w k H ( A ) = 0 . Consequently, by Proposition 2.4, T ( w ) ∈ C ( A ) and lim t → T ∗ k T ( η ( t, u )) − T ( w ) k C ( A ) = 0 . From this, by differentiating e t η ( t, u ), we deduce that η ( t, u ) = e − t (cid:16) u + Z t e s T ( η ( s, u )) ds (cid:17) → e − T ∗ (cid:16) u + Z T ∗ e s T ( η ( s, u )) ds (cid:17) in C ( A ) as t → T ∗ . By uniqueness of the limit we have that the right hand side above coincides with w and a posteriori it follows that η ( t, u ) → w in C ( A ) as t → T ∗ . This contradicts(3.7), hence it follows that T ∗ = ∞ .To show the existence of a sequence { w n } n with the asserted properties, we argueby contradiction again and assume that there exists t , δ > k Φ( η ( t, u )) k H ( A ) ≥ δ for t ≥ t .By (3.3), we then deduce that I ( η ( t , u )) − I ( η ( t, u )) ≥ ( t − t ) δ → ∞ as t → T ∗ = ∞ , which contradicts assumption (3.4). Hence there exists a sequence { s n } n ⊂ (0 , + ∞ )with the required properties. (cid:3) Given Lemmas 3.1 and 3.5, it only remains to exhibit u ∈ K satisfying (3.4) andthe additional condition that the related Palais-Smale sequence does not convergeto zero. This is the content of the next subsection.3.3. A dynamical systems point of view.
Partially inspired by [2], we showthat the mountain pass geometry of the functional I allows to construct a subsetof K that is invariant for the flow and with the property that I is strictly positiveover this set, see Lemma 3.9 below. This set is defined as the boundary of a certaindomain of attraction for the flow η .Let α be given as in Lemma 3.2. We define L α := { u ∈ B α ( K ) : I ( u ) < ρ α } . It is not difficult to check that L α is relatively open in K with respect to the C -norm, that is to say, for every u ∈ L α there exists ε > { v ∈ C ( A ) : k v − u k C ( A ) < ε } ∩ K ⊂ L α . (3.8)Moreover, L α has the following positive invariance property. Lemma 3.6.
For u ∈ L α , we have T max ( u ) = ∞ and η ( t, u ) ∈ L α for all t ≥ .Proof. By Lemma 3.4, we know that η ( t, u ) ∈ K for all t ∈ (0 , T max ( u )). Supposeby contradiction that there exists t ∈ (0 , T max ( u )) such that η ( t , u ) L α . Since,by Lemma 3.5, I ( η ( t , u )) ≤ I ( u ) < ρ α , necessarily k η ( t , u ) k H ( A ) ≥ α . We ob-serve that the map t ∈ [0 , T max ( u ))
7→ k η ( t, u ) k H ( A ) is continuous, by virtue of thecontinuous embedding C ( A ) ֒ → H ( A ), therefore there exists t ∈ (0 , t ] such that η ( t , u ) ∈ S α ( K ). This contradicts Lemma 3.2(ii), being I ( η ( t , u )) < ρ α . Conse-quently, η ( t, u ) ∈ L α for all t ∈ (0 , T max ( u )), and therefore lim t → T max ( u ) I ( η ( t, u )) ≥ T max ( u ) = ∞ by Lemma 3.5. (cid:3) Next we consider the domain of attraction of L α in K , more precisely D ( L α ) := { u ∈ K : η ( t, u ) ∈ L α for some t ∈ (0 , T max ( u )) } . We notice that Lemma 3.6 implies thatif u ∈ D ( L α ) then T max ( u ) = ∞ and η ( t, u ) ∈ D ( L α ) for all t ≥ . (3.9)Moreover, Lemmas 3.2 (i), 3.5 and 3.6 provideinf u ∈ D ( L α ) I ( u ) ≥ . (3.10) Lemma 3.7. D ( L α ) is relatively open in K with respect to the C -norm, that is tosay, for every u ∈ D ( L α ) there exists δ > such that { v ∈ C ( A ) : k v − u k C ( A ) < δ } ∩ K ⊂ D ( L α ) . Proof.
Let u ∈ D ( L α ). By definition there exists t ∈ [0 , T max ( u )) such that η ( t , u ) ∈ L α . On the one hand, being L α relatively open in K with respect tothe C -norm (see (3.8)), there exists ε > { w ∈ C ( A ) : k w − η ( t , u ) k C ( A ) < ε } ∩ K ⊂ L α . (3.11)On the other hand, given such ε >
0, by Remark 3.3 there exists δ > v ∈ C ( A ) , k v − u k C ( A ) < δ implies k η ( t , v ) − η ( t , u ) k C ( A ) < ε. (3.12)By combining (3.11) and (3.12), we deduce that such δ > (cid:3) We denote by Z α the relative boundary of D ( L α ) in K with respect to the C -norm. In view of Lemma 3.7 and of the fact that K is closed with respect to the C -topology, we have more explicitly Z α := D ( L α ) \ D ( L α ) , (3.13)where D ( L α ) denotes the standard closure of D ( L α ) in C ( A ) with respect to the C -norm. Lemma 3.8.
The set Z α defined in (3.13) is not empty. More precisely, for every ψ ∈ K \ { } there exists t ∗ > such that t ∗ ψ ∈ Z α .Proof. For ψ ∈ K \ { } , let J ψ := { t ≥ tψ ∈ D ( L α ) } . On the one hand, there exists ε > , ε ) ⊂ J ψ because 0 ∈ L α ⊂ D ( L α )and D ( L α ) is relatively open in K in virtue of Lemma 3.7. On the other hand, J ψ is bounded, as there exists ¯ t > I ( tψ ) ≤ − t ≥ ¯ t , whichimplies that tψ D ( L α ) for every t ≥ ¯ t by virtue of (3.10). As a consequence, wehave that t ∗ := sup J ψ ∈ (0 , ∞ ) . Then t ∗ ψ ∈ Z α , by definition of Z α . (cid:3) By the continuity of the flow η with respect to the C -norm (see Remark 3.3),the following property is a consequence of Lemmas 3.5 and 3.6. Lemma 3.9.
For u ∈ Z α , we have T max ( u ) = ∞ and η ( t, u ) ∈ Z α , I ( η ( t, u )) ≥ ρ α for all t ≥ . Proof.
First we notice that, for u ∈ Z α , T max ( u ) = ∞ by virtue of Lemma 3.5 (seein particular condition (3.4)) and of property (3.10).Next we prove that, if u ∈ Z α , η ( t, u ) ∈ Z α for every t >
0. To this aim, supposeby contradiction that there exists t > η ( t , u ) ∈ D ( L α ) ∪ ( K \ D ( L α )).If η ( t , u ) ∈ D ( L α ), by definition of D ( L α ), there exists t ∈ ( t , T max ( u )) such that η ( t , u ) ∈ L α . This means that u ∈ D ( L α ), which is impossible by definition of Z α .It remains to rule out the possibility that η ( t , u ) ∈ K \ D ( L α ). Being K \ D ( L α )relatively open in K , there exists ε = ε ( t ) > v ∈ K , k v − η ( t , u ) k C ( A ) < ε implies v ∈ K \ D ( L α ) . (3.14)Now, since u ∈ Z α , there exists a sequence { v n } n with the property that v n ∈ D ( L α ) for every n ∈ N , lim n → + ∞ k v n − u k C ( A ) = 0 . (3.15)Therefore, by Remark 3.3, given ε as in (3.14), there exists n ∈ N such that k η ( t , v n ) − η ( t , u ) k C ( A ) < ε for every n ≥ n . (3.16)By combining (3.14) and (3.16), we infer that η ( t , v n ) ∈ K \ D ( L α ) for every n ≥ n . (3.17)On the other hand, since { v n } ⊂ D ( L α ), η ( t , v n ) ∈ D ( L α ) for every n (see (3.9)).This contradicts (3.17) and concludes this part of the proof. Let us prove the third property, that is to say, if u ∈ Z α then I ( η ( t, u )) ≥ ρ α forall t ≥
0. We proceed again by contradiction. Let ¯ t ≥ I ( η (¯ t, u )) < ρ α (3.18)Being u D ( L α ), we deduce that k η (¯ t, u ) k H ( A ) ≥ α . From the definition of ρ α weinfer that indeed k η (¯ t, u ) k H ( A ) > α. (3.19)Now, let { v n } be as in (3.15). On the one hand, (3.19) and the continuous depen-dence of η on the initial data (see Remark 3.3) imply the existence of ¯ n ∈ N suchthat k η (¯ t, v n ) k H ( A ) > α for every n ≥ ¯ n. (3.20)On the other hand, since { v n } ⊂ D ( L α ) for every n ∈ N , there exists a sequence { t n } ⊂ [0 , + ∞ ) such that k η ( t n , v n ) k H ( A ) < α and I ( η ( t n , v n )) < ρ α , for every n ∈ N . Then Lemma 3.6 provides k η ( t, v n ) k H ( A ) < α and I ( η ( t, v n )) < ρ α , for every t ≥ t n , n ∈ N . (3.21)From (3.20) and (3.21) we deduce that ¯ t < t n for every n ∈ N , and that, for every n ≥ ¯ n there exists s n ∈ (¯ t, t n ) such that k η ( s n , v n ) k H ( A ) = α for every n ≥ ¯ n . Bydefinition of ρ α , we have I ( η ( s n , v n )) ≥ ρ α for every n ≥ ¯ n . Being s n ≥ ¯ t , Lemma3.5 provides I ( η (¯ t, v n )) ≥ ρ α for every n ≥ ¯ n . Passing to the limit (see Remark3.3) we infer that I ( η (¯ t, u )) ≥ ρ α , which contradicts (3.18). (cid:3) Proof of Theorem 1.1.
Proof of Theorem 1.1.
Let ψ ∈ K \ { } and let u := t ∗ ψ ∈ Z α , with t ∗ as in Lemma3.8. By Lemma 3.5 and Lemma 3.9 we have that T max ( u ) = ∞ and that thereexists a sequence { s n } n ⊂ (0 , + ∞ ) such that lim n → + ∞ s n = + ∞ andlim n → + ∞ k Φ( w n ) k H ( A ) = 0 , (3.22)for the sequence { w n } n defined in (3.6). By Lemma 3.1, we may pass to a subse-quence such that w n → w in H ( A ) for some w ∈ K and T ( w ) = w . Lemma 3.9provides k w k H ( A ) = lim n → + ∞ k w n k H ( A ) ≥ n → + ∞ I ( w n ) ≥ ρ α , (3.23)thus implying that w is nontrivial. Consequently, w is a nontrivial solution of (1.2)belonging to N K ⊂ K .Next, we assume in addition that the function ψ above satisfies, in addition, that ψ ∈ N K and I ( ψ ) = c I . Here c I is defined in (1.8), so ψ is a minimizer of I on N K . In this case the function w ∈ N K found above satisfies c I ≤ I ( w ) ≤ I ( t ∗ ψ ) ≤ I ( ψ ) = c I , (3.24)where in the first inequality we used that w ∈ N K and in the third we used thatsup t> I ( tψ ) = I (1 ψ ), being ψ ∈ N K , cf. (2.16). As for the second inequality,since w = lim n w n in H ( A ), k w n k L p ( A ) → k w k L p ( A ) by Lemma 2.3. By theproperties of a , the norm ( R A a ( x ) |·| p dx ) /p is equivalent to k ·k L p ( A ) , hence I ( w ) =lim n I ( w n ). Thus, the second inequality in (3.24) is obtained recalling that I ( w n ) = I ( η ( s n , t ∗ ψ )) ≤ I ( t ∗ ψ ) for every n and passing to the limit in n . So, equality holdsin all of the inequalities in (3.24). In particular, being I ( t ∗ ψ ) = I ( ψ ) and ψ ∈ N K ,we obtain t ∗ = 1. Hence, I ( w n ) = I ( η ( s n , ψ )) ≤ I ( ψ ) = c I for every n . On theother hand, by (3.3), I ( η ( s n , ψ )) ց c I . Therefore, I ( η ( s n , ψ )) = c I for every n ,and so, by the monotonicity of I ( η ( · , ψ )) and since lim n s n = + ∞ , it follows that I ( η ( t, ψ )) = c I for all t ∈ (0 , ∞ ). Therefore, by (3.3),Φ( η ( t, ψ )) = 0 for all t ∈ (0 , ∞ ).Consequently, T ( η ( t, ψ )) = η ( t, ψ ) for all t ∈ (0 , ∞ ). Passing to the limit t + and using the continuity of T , we deduce that T ( ψ ) = ψ , hence ψ is a nontrivialsolution of (1.2) belonging to K .To finish the proof of Theorem 1.1, we still have to show that the minimal value c I of the functional I is positive and attained on the set N K . For this we let { ψ ℓ } ℓ be a sequence in N K with the property that I ( ψ ℓ ) → c I as ℓ → ∞ .We let t ℓ ∗ be given as in Lemma 3.8 corresponding to ψ ℓ . Repeating the argu-ment above for every ℓ yields corresponding nontrivial solutions w ℓ ∈ N K of (1.2)satisfying c I ≤ I ( w ℓ ) ≤ I ( t ℓ ∗ ψ ℓ ) ≤ I ( ψ ℓ ) = c I + o (1) as ℓ → ∞ .Notice that, by (3.23) with w = w ℓ , we know k w ℓ k H ( A ) ≥ ρ α , for every ℓ. (3.25)Since c I + o (1) = I ( w ℓ ) = I ( w ℓ ) − p I ′ ( w ℓ ) w ℓ = (cid:16) − p (cid:17) k w ℓ k H ( A ) as ℓ → ∞ , the sequence { w ℓ } ℓ is bounded in H ( A ). Passing to a subsequence, we may assumethat w ℓ ⇀ ¯ w in H ( A ). Since e K is weakly closed (see Lemma 2.2), we have ¯ w ∈ e K .Moreover, by Proposition 2.4, we have k w ℓ − T ( ¯ w ) k H ( A ) = k T ( w ℓ ) − T ( ¯ w ) k H ( A ) → ℓ → ∞ ,so w ℓ → T ( ¯ w ) strongly in H ( A ) as ℓ → ∞ . By uniqueness of the weak limit,¯ w = T ( ¯ w ), and therefore w ℓ → ¯ w strongly in H ( A ). From this we deduce, byProposition 2.4 that ¯ w ∈ K and that w ℓ → ¯ w in C ( A ) as ℓ → ∞ .Consequently, ¯ w is a critical point of I with I ( ¯ w ) = lim ℓ →∞ I ( w ℓ ) = c I >
0, the lastinequality coming from (3.25). In particular, ¯ w
0, so ¯ w ∈ N K . Hence the minimalvalue c I is attained by the functional I in N K . (cid:3) Remark . Notice that the existence of a nontrivial solution of (1.2) followsalready from (3.23). The remaining part of the proof of Theorem 1.1 gives a vari-ational characterization that will be useful in the next section to prove the non-radiality of the solution when a is constant and some additional assumptions on p or A hold. The case of constant a In this section we treat problem (1.9) where the weight function a in (1.2) satisfies a ≡
1. We recall that, for every fixed p >
2, (1.9) admits a unique positive radialsolution u rad ∈ C ( A ) by [27]. We continue using the notation introduced in theprevious sections in the special case a ≡
1. In the next proposition we collect someproperties satisfied by u rad which will be useful in the sequel. Proposition 4.1.
Let P := { u ∈ C ( A ) : u ≥ } . The radial solution u rad belongs to the interior of P with respect to the C -norm. Moreover, the followinginequalities hold I ( u rad ) ≥ I ( tu rad ) for every t ≥ and I ′ (cid:0) tu rad (cid:1) u rad > > I ′ (cid:0) t ′ u rad (cid:1) u rad for every t ∈ (0 , and t ′ ∈ (1 , ∞ ) . (4.2) Proof.
Clearly u rad ∈ P , moreover, by the Hopf Lemma, u rad is contained in theinterior of P with respect to the C -norm. Now, since u rad is a solution of (1.9), u rad ∈ N K and so t u rad = 1, cf. (2.16). Thus, the function t ∈ [0 , ∞ ) I ( tu rad )admits a unique maximum in t = 1, and so (4.1) follows. Moreover, the samefunction t I ( tu rad ) is strictly increasing in (0 ,
1) and strictly decreasing in (1 , ∞ ),this implies (4.2) and concludes the proof. (cid:3) Our main tool to prove the existence of nonradial solutions of (1.9) will be thefollowing criterion related to instability with respect to specific directions.
Proposition 4.2.
Suppose that there exists an axially symmetric function v ∈ C ( A ) , written in polar coordinates as v = v ( r, θ ) , satisfying the following proper-ties: I ′′ ( u rad )( v, v ) <
0; (4.3) ∂ θ v ( r, θ ) ≤ for ( r, θ ) ∈ ( R , R ) × (0 , π/ v ( r, θ ) = v ( r, − θ ) for ( r, θ ) ∈ ( R , R ) × (0 , π/ ; (4.5) Z S N − v ( r, · ) dσ = 0 for every r ∈ ( R , R ) , (4.6) where, in the last relation, the two-variable function v ( r, θ ) is meant as an N -variable function v ( r, θ, φ , . . . , φ N − ) which is constant with respect to φ , . . . , φ N − .Then we have c I < I ( u rad ) , (4.7) so every minimizer u ∈ N K of I (cid:12)(cid:12)(cid:12) N K is nonradial.Proof. By assumption (4.3) and the continuity of I ′′ , there exist δ ∈ (0 ,
1) and ρ > I ′′ (cid:0) t ( u rad + τ v ) (cid:1) ( v, v ) < t ∈ [1 − δ, δ ], τ ∈ [ − ρ, ρ ]. (4.8)Since, by Proposition 4.1, u rad is contained in the interior of P with respect to the C -norm, we may also assume, by adjusting δ and ρ if necessary, that t ( u rad + τ v ) ≥ A for t ∈ [1 − δ, δ ], τ ∈ [ − ρ, ρ ].Combining this information with assumptions (4.4) and (4.5), we deduce that t ( u rad + τ v ) ∈ e K for t ∈ [1 − δ, δ ], τ ∈ [ − ρ, ρ ]. Moreover, since, by (4.2), I ′ (cid:0) (1 − δ ) u rad (cid:1) u rad > > I ′ (cid:0) (1 + δ ) u rad (cid:1) u rad , there exists s ∈ (0 , ρ ) with I ′ (cid:0) (1 − δ )( u rad + sv ) (cid:1) ( u rad + sv ) > > I ′ (cid:0) (1 + δ )( u rad + sv ) (cid:1) ( u rad + sv )By the intermediate value theorem, there exists t ∈ [1 − δ, δ ] with I ′ (cid:0) t ( u rad + sv ) (cid:1) ( u rad + sv ) = 0 and therefore u ∗ := t ( u rad + sv ) ∈ N K . Moreover, since u rad ∈ N K , by a Taylor expansion, (4.1) and (4.8) we have I ( u ∗ ) − I ( u rad ) ≤ I ( u ∗ ) − I ( tu rad )= stI ′ ( t u rad ) v + t Z s I ′′ ( t ( u rad + τ v ))( v, v )( s − τ ) dτ< stI ′ ( t u rad ) v = st (cid:18) t h u rad , v i H ( A ) − t p − Z A u p − v dx (cid:19) = st (1 − t p − ) Z R R r N − u p − ( r ) Z S N − v ( r, · ) dσdr = 0where we used assumption (4.6) in the last step. Consequently, c I ≤ I ( u ∗ )
Let α be the first eigenvalue of the one dimensional weighted eigen-value problem − w rr − N − r w r + (cid:0) − ( p − u p − (cid:1) w = αr w in ( R , R ) , w ( R ) = w ( R ) = 0 , (4.9) and let w be the (up to normalization) unique positive corresponding eigenfunction.Let then Y ( θ ) := 1 − N sin θ , θ ∈ ( − π/ , π/ be the (up to sign and normalization)unique axially symmetric spherical harmonic of degree two.If α + 2 N < then v = v ( r, θ ) = w ( r ) Y ( θ ) satisfies assumptions (4.3) – (4.6) inProposition 4.2.Proof. By construction, v ∈ C ( A ) and satisfies assumptions (4.4) and (4.5) ofProposition 4.2. Moreover, Z S N − v ( r, · ) dσ = w ( r ) Z S N − (1 − N x N ) dσ ( x ) = w ( r ) Z S N − (1 − | x | ) dσ ( x ) = 0for every r ∈ ( R , R ), so assumption (4.6) is also satisfied. It remains to prove(4.3). To this aim, we recall that the function Y is an eigenfunction of the Laplace-Beltrami operator − ∆ S N − on the unit sphere S N − corresponding to the eigenvalue λ = 2 N . By using (2.2), it is straightforward to verify that − ∆ v + v − ( p − u p − v = α + 2 N | x | v in A. By testing this equation by v and integrating by parts, we obtain Z A (cid:16) |∇ v | + v − ( p − u p − v (cid:17) dx = ( α + 2 N ) Z A v | x | dx < , by assumption. Since the left hand side is I ′′ ( u rad )( v, v ), the proof is concluded. (cid:3) Proof of Theorem 1.2 (completed).
By combining Proposition 4.2 and Lemma 4.3,it remains to prove that there exists p ∗ > α < − N for every p > p ∗ .This comes from the fact that the first eigenvalue α of the eigenvalue problem (4.9)satisfies, as a function of the exponent p >
2, the asymptotic expansion α = α ( p ) = − cp + o ( p ) as p → ∞ with a constant c > , as proved in [20, Proposition 4.5]. Hence every K -ground state solution of (1.9) isnonradial for p > p ∗ , thus concluding the proof. (cid:3) Proof of Theorem 1.3 (completed).
Here we consider p > A = A R , i.e., R = R , R = R + 1 with parameter R >
0. Similarly as before,it only remains to prove that there exists an R ∗ such that α < − N for R > R ∗ By [20, Proposition 3.2], the first eigenvalue α of the eigenvalue problem (4.9)satisfies, as a function of the inner radius R >
0, the asymptotic expansion α = α ( R ) = − cR + o ( R ) as R → ∞ with a constant c > . Hence every K -ground state solution of (1.9) is nonradial for R > R ∗ , thus con-cluding the proof. (cid:3) Acknowledgments
The authors acknowledge the support of the Departement of Mathematics of theUniversity of Turin. A. Boscaggin, F. Colasuonno and B. Noris were partially sup-ported by the INdAM - GNAMPA Project 2019 “Il modello di Born-Infeld perl’elettromagnetismo nonlineare: esistenza, regolarit`a e molteplicit`a di soluzioni”and by the INdAM - GNAMPA Project 2020 “Problemi ai limiti per l’equazionedella curvatura media prescritta”. B. Noris acknowledges the support of the pro-gram S2R of the Universit´e de Picardie Jules Verne, which financed a short visit toTurin, where part of this work has been achieved.
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Alberto BoscagginDipartimento di MatematicaUniversit`a di Torinovia Carlo Alberto 10, 10123 Torino, Italy
Email address : [email protected] Francesca ColasuonnoDipartimento di MatematicaUniversit`a di Bolognap.zza di Porta San Donato 5, 40126 Bologna, Italy
Email address : [email protected] Benedetta NorisDipartimento di MatematicaPolitecnico di Milanop.zza Leonardo da Vinci 32, 20133 Milano, Italy
Email address : [email protected] Tobias WethInstitut f¨ur MathematikGoethe-Universit¨at FrankfurtRobert-Mayer-Str. 10, D-60629 Frankfurt am Main, Germany
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