Regularity and Symmetry for Semilinear Elliptic Equations in Bounded Domains
aa r X i v : . [ m a t h . A P ] F e b REGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS INBOUNDED DOMAINS.
LOUIS DUPAIGNE AND ALBERTO FARINA
Résumé.
In the present paper, we investigate the regularity and symmetry properties of weak solutionsto semilinear elliptic equations which are locally stable. Introduction and main results
In the present paper, we investigate the regularity and symmetry properties of weak solutions tosemilinear elliptic equations. We shall focus on the following class :
Definition 1.
Let N ≥ , Ω ⊂ R N denote an open set and f ∈ C ( R ) . Assume that u ∈ H loc (Ω) , f ( u ) ∈ L loc (Ω) and that u solves (1) − ∆ u = f ( u ) in D ′ (Ω) . We say that u is locally stable in Ω if f ′ ( u ) ∈ L loc (Ω) and if for every x ∈ Ω , there exists an openneighborhood ω ⊂ Ω of x such that for every ϕ ∈ C c ( ω ) , there holds (2) ˆ ω f ′ ( u ) ϕ ≤ ˆ ω |∇ ϕ | . A solution is stable in Ω if the above inequality holds for ω = Ω and for every ϕ ∈ C c (Ω) . As shown by the following examples, the class of locally stable solutions is natural and wide enoughto encompass various interesting families of solutions (naturally) arising in the study of PDEs.(1) Smooth solutions are locally stable, thanks to the (sharp) Poincaré inequality.(2) More generally, for N ≥ , weak solutions such that f ′ ( u ) ∈ L loc (Ω) and f ′ ( u ) + ∈ L N/ loc (Ω) arelocally stable. Indeed, choosing ω so small that k f ′ ( u ) + k L N ( ω ) ≤ N ( N − | B | and applying Hölder’sand Sobolev’s inequalities, we have ˆ ω f ′ ( u ) ϕ ≤ ˆ ω f ′ ( u ) + ϕ ≤ k f ′ ( u ) + k L N ( ω ) k ϕ k L NN − ( ω ) ≤ ˆ ω |∇ ϕ | When N = 2 , the local stability follows from Moser-Trudinger inequality if f ′ ( u ) ∈ L loc (Ω) and f ′ ( u ) + ∈ L ploc (Ω) for some p > .(3) If N > , f ( u ) = 2( N − e u and u = − | x | , then f ′ ( u ) = N − | x | ∈ L loc but just fails to belongto L N/ near the origin. By the optimality of Hardy’s inequality, u is never locally stable in anyopen set containing the origin whenever ≤ N ≤ .(4) Local minimizers are stable : u ∈ H loc (Ω) is a local minimizer if for any Ω ′ ⊂⊂ Ω and for all ϕ ∈ C c (Ω ′ ) , t = 0 is a point of minimum of the function t e ( t ) := E Ω ′ ( u + tϕ ) , where E Ω ′ ( v ) = ´ Ω ′ (cid:0) |∇ v | − F ( v ) (cid:1) and F ′ = f . Therefore (2) holds (since e ′′ (0) ≥ .(5) If u ∈ H loc (Ω) has finite Morse index in Ω , then u is locally stable in Ω , see Proposition 1.5.1 in[10] (or Proposition 2.1 in [9]). In addition, u is stable outside a compact set, see Remark 1 in [12].But there are also locally stable solutions of infinite Morse index. This is the case e.g. when Ω isthe punctured unit ball, f ( u ) = 2( N − e u , u ( x ) = − | x | and ≤ N ≤ .
1. We recall that a solution u to (1) has Morse index equal to K ≥ , if f ′ ( u ) ∈ L loc (Ω) and K is the maximal dimensionof a subspace X K of C c (Ω) such that ´ Ω |∇ ψ | < ´ Ω f ′ ( u ) ψ for any ψ ∈ X K \ { } . In particular u is stable if and only ifits Morse index is zero. Our first result concerns the complete classification of nonnegative stable solutions u ∈ H (Ω) to (1),when f is a convex function satisfying f (0) = 0 . Theorem 1.
Let Ω be a bounded domain of R N , N > , let f ∈ C ( R ) be a convex function such that f (0) = 0 and let λ be the principal eigenvalue of − ∆ with homogeneous Dirichlet boundary conditions.Assume that u ∈ H (Ω) , f ( u ) ∈ L loc (Ω) and that u is a stable solution to (3) ( − ∆ u = f ( u ) in D ′ (Ω) u > a.e. on Ω . Then, either u ≡ or f ( t ) = λ t on (0 , sup Ω u ) and u ∈ C ∞ (Ω) ∩ H (Ω) is a positive first eigenfunctionof − ∆ with homogeneous Dirichlet boundary conditions.Remark . If u ≡ , then necessarily f ′ (0) λ , by Lemma 14 in section 3 below. Also observe that forany α λ there is a convex function f satisfying f (0) = 0 , f ′ (0) = α and such that u ≡ is a stablesolution to (3). An example is provided by f ( u ) = u + αu .The latter result is a consequence of the following general theorem which holds true for any convexfunction f of class C and for distributional solutions merely in H (Ω) . Theorem 3.
Let Ω be a bounded domain of R N , N > and let f ∈ C ([0; + ∞ )) be a convex function.Assume that u, v ∈ H (Ω) satisfy u − v ∈ H (Ω) , v u a.e. on Ω , f ( u ) , f ( v ) ∈ L loc (Ω) and both u and v are solution to (4) − ∆ w = f ( w ) in D ′ (Ω) . If f ′ ( u ) ∈ L loc (Ω) and u is stable, then either u ≡ v or f ( t ) = a + λ t for all t ∈ (inf Ω v, sup Ω u ) andsome a ∈ R , u, v ∈ C ∞ (Ω) and u − v is a positive first eigenfunction of − ∆ with homogeneous Dirichletboundary conditions.Remark . According to Theorem 1.3 and Corollary 3.7 in [7], there exists a C , positive, increasing butnon-convex nonlinearity f with two distinct and ordered (classical) stable solutions ≤ u ≤ v . In otherwords, the convexity assumption cannot be completely removed from the above theorem.Another important consequence of Theorem 3 is the following approximation result which, in turn,motivated our definition of local stability (see Definition 1). This result will be central in the proof of ourmain regularity results for locally stable solutions to (1). Theorem 5.
Assume α ∈ (0 , and N > .Let Ω be a bounded domain of R N and let f ∈ C ([0 , + ∞ )) be a convex function such that f (0) > .Assume that u ∈ H (Ω) , f ( u ) ∈ L loc (Ω) and that u is a stable solution to (5) ( − ∆ u = f ( u ) in D ′ (Ω) u > a.e. on Ω . Then, there is a nondecreasing sequence ( f k ) of convex functions in C ([0 , + ∞ )) ∩ C , ([0 , + ∞ )) suchthat f k ր f pointwise in [0; + ∞ ) and a nondecreasing sequence ( u k ) of functions in H (Ω) ∩ C (Ω) such
2. Actually the real number a is unique and its value is given by − λ ´ Ω φ h ´ Ω φ , where φ is a positive first eigenfunctionof − ∆ with homogeneous Dirichlet boundary conditions and h ∈ H (Ω) is the unique weak solution of − ∆ h = 0 in Ω with u − h ∈ H (Ω) (to see this, use φ as test function in the weak formulation of − ∆ u = a + λ u and the fact that h isharmonic) and also note that h is nonnegative by the maximum principle. In particular, u, v ∈ H (Ω) ⇐⇒ a = 0 . Also notethat, for every a there exist solutions u, v for which the second alternative of the theorem occurs. Indeed, the functions u t := − aλ + tφ , t > are suitable.3. Indeed, applying Corollary 3.7 in [7], we see that in the notations of that corollary, for f ( u ) = λg ( u ) , u λ is minimalhence stable. In addition, by minimality, λ u λ is nondecreasing and so u λ converges to a stable solution v as λ ց λ suchthat v ≥ u λ . Then, take u = u λ EGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS. 3 that u k is a stable weak solution to (6) − ∆ u k = f k ( u k ) in Ω ,u k − u ∈ H (Ω) , u k u a.e. on Ω , and (7) u k −→ u in H (Ω) , u k ր u a.e. on Ω . Moreover, if f is nonnegative, then any function f k is nonnegative too.Remark . (1) It follows from (7) that under the assumptions of the proposition, locally stable solutionsare automatically lower semi-continuous.(2) The proposition recovers and extends Corollary 3.2.1. in [10].(3) The result is not true if we drop the assumption u ∈ H (Ω) . To see this, consider Example 3.2.1 in[10] in the light of Theorem 7 below.(4) We do not know if the assumption f convex can be dropped.(5) Theorem 5 generalizes Proposition 21 below, in which the approximating nonlinearity is taken of theform f k = (1 − ǫ k ) f , with ǫ k → at the expense of additionally assuming that f is nondecreasing.Theorem 5 can be combined with the following a priori estimate due to [5] in order to establishsmoothness of locally stable solutions when f is nonnegative, convex and N ≤ . Theorem A ([5]) . Let B be the unit ball of R N , N > . Assume that u ∈ C ( B ) is a stable solutionof (1) in Ω = B , where f : R → R is locally Lipschitz and nonnegative. If ≤ N ≤ , then (8) k u k C α ( B / ) ≤ C k u k L ( B ) , where α ∈ (0 , , C > are dimensional constants. More precisely we have the following interior regularity result :
Theorem 7.
Let f ∈ C ([0 , + ∞ )) be a nonnegative convex function. Let Ω be an open set of R N , N > .Assume that u ∈ H loc (Ω) , f ( u ) ∈ L loc (Ω) and that u is a locally stable solution of (1) such that u ≥ a.e. in Ω .If N , then u ∈ C ,βloc (Ω) for all β ∈ (0 , . In particular, any finite Morse index solution is smoothin Ω .Remark . The result is optimal since for N ≥ , f ( u ) = 2( N − e u and Ω = B , u ( x ) = − | x | is asingular stable solution in H ( B ) . Also observe that the above theorem fails if we do not assume that u belongs to H loc (Ω) , see e.g. Example 3.2.1 in [10].A priori estimates near the boundary are more subtle, as the following result shows. Theorem 9.
Let f ∈ C ([0 , + ∞ )) be a nonnegative convex function. Let Ω be an open set of R N , N > .Assume that u ∈ H (Ω) , f ( u ) ∈ L loc (Ω) and that u is a finite Morse index solution of (1) such that u ≥ a.e. in Ω .(1) Let Ω be a bounded uniformly convex domain of class C ,α , for some α ∈ (0 , . Then there existsconstants ρ, γ > , depending only on Ω , such that (9) k u k L ∞ (Ω ρ ) γ k u k L (Ω) where Ω ρ := { x ∈ Ω : dist ( x, ∂ Ω) < ρ } . In particular, u ∈ C ,α (Ω ρ ∪ ∂ Ω) .(2) If ≤ N ≤ and either Ω is C ,α and convex or f is nondecreasing and Ω is C , then there existsconstants ρ, γ > , depending only on Ω , such that (9) holds.
4. That is a function u k satisfying ´ Ω ∇ u k ∇ ϕ = ´ Ω f k ( u k ) ϕ , for all ϕ ∈ H (Ω) . LOUIS DUPAIGNE AND ALBERTO FARINA (3) Fix N ≥ . For every sequence ( ρ n ) ⊂ R ∗ + converging to zero, there exists a sequence of bounded C convex domains Ω n ⊂ R N , n ∈ N ∗ , such that the corresponding stable solution u n to (1) with f ( u ) = 2( N − e u and Ω = Ω n satisfies k u n k L ∞ ((Ω n ) ρn ) → + ∞ yet | Ω n | k u n k L (Ω n ) remains bounded. The last point of the above theorem shows that in dimension N ≥ , no universal a priori estimateof the type (9) can hold near the boundary if the domain Ω is merely convex and the constant γ dependson the dimension N and the volume | Ω | only. The case N = 10 is open. We do not know either if locallystable solutions are smooth near the boundary of convex domains, although the universal a priori estimatefails.When Ω is bounded and rotationally invariant we can prove the following classification result. Theorem 10. (1) Let
R > , N ≥ , B be the open ball B (0 , R ) ⊂ R N and let f ∈ C ([0 , + ∞ )) be aconvex function. Assume that u ∈ H ( B ) , f ( u ) ∈ L loc ( B ) and that u is a stable solution to (10) ( − ∆ u = f ( u ) in D ′ ( B ) u > a.e. on B .Then, either u ≡ or u ∈ C ( B \ { } ) , u > and u is radially symmetric and radially strictlydecreasing. Furthermore, if N and f is nonnegative, then u ∈ C ( B ) .(2) Let R > , N , B be the open ball B (0 , R ) ⊂ R N and let f ∈ C ([0 , + ∞ )) be a nonnegativeconvex function. Assume that u ∈ H ( B ) , f ( u ) ∈ L loc ( B ) and that u solves (11) ( − ∆ u = f ( u ) in D ′ ( B ) u > a.e. on B. If u has finite Morse index, then either u ≡ or u ∈ C ( B ) , u > and u is radially symmetricand radially strictly decreasing.(3) Let N ≥ , Ω ⊂ R N be an open annulus centered at the origin and let f ∈ C ([0 , + ∞ )) be a convexfunction. Assume that u ∈ H (Ω) , f ( u ) ∈ L loc (Ω) and that u is a stable solution to (12) ( − ∆ u = f ( u ) in D ′ (Ω) u > a.e. on Ω . Then, either u ≡ or u ∈ C (Ω) , u > and u is radially symmetric. Remark . (1) The conclusion that u is radially symmetric in item (1) of Theorem 10 was alreadyknown to hold true in the special case where u ∈ C ( B ) (and with no additional sign assumptionon u ), see e.g. [1].(2) Item (2) of Theorem 10 is sharp. Indeed, for N ≥ , f ( u ) = 2( N − e u and Ω = B = B (0 , , u ( x ) = − | x | is a singular stable solution in H ( B ) . The finite Morse index assumption is alsoessential : for N = 3 and f ( u ) = 2 e u , there exists a family of nonradial singular solutions in Ω = B of the form u ( x ) = − | x − x | + v ( x ) , where x = 0 and v ∈ L ∞ ( B ) ∩ H ( B ) , see (in details)the proofs in [18]. In particular, u ∈ H ( B ) . It follows from our result that u cannot have finiteMorse index. In contrast, note that, for N , Ω = B and f ( u ) = 2( N − e u , there existinfinitely many smooth and positive solutions to (1) such that u = 0 on ∂ Ω and with finite andnon-zero Morse index. See chapter 2 of [10] (and the references therein) for a detailed discussion ofthis topic.(3) It will be clear from the proof that :(a) the radial symmetry in item (1) is still true if we replace u ∈ H (Ω) by any member u of H (Ω) having constant trace c > on ∂B .
5. Furtheremore, if Ω is the annulus { x ∈ R N : 0 < a < | x | < b } and u ( x ) = v ( | x | ) , then there is a unique r ∈ ( a, b ) such that v ′ > in ( a, r ) , v ′ ( r ) = 0 and v ′ < in ( r , b ) . The result follows as in the proof of item 1). For this reason weomit it.
EGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS. 5 (b) the radial symmetry in item (3) is still true if we replace u ∈ H (Ω) by any member u of H (Ω) having a nonnegative constant trace on each of the two connected components of the boundaryof the annulus (possibly with differents values on the two connected component).(c) Note that, if c > , we do not claim any monotonicity or special property about the radialprofile (as it happens when c = 0 .)A crucial step in the proof of the above Theorem is the following general symmetry result. As we shallsee below, this result also enables us to prove further symmetry results for stable solutions in "symmetric"bounded domains. Proposition 12.
Assume N > and let f ∈ C ([0 , + ∞ )) be a convex function. Let ρ ∈ O ( N ) and let Ω ⊂ R N be a ρ − invariant bounded domain, i.e., a bounded domain such that ρ (Ω) = Ω . Assume that u ∈ H c (Ω) , f ( u ) ∈ L loc (Ω) and that u is a stable solution to (13) ( − ∆ u = f ( u ) in D ′ (Ω) u > a.e. on Ω .If ρ has a fixed point in Ω , then u is ρ -invariant, namely, u ( x ) = u ( ρx ) for almost every x ∈ Ω .Remark . Below we provide a (non-exhaustive) list of bounded domains to which the above resultapplies :(1) if Ω is any bounded domain symmetric with respect to a hyperplane, then u inherits the samesymmetry.(2) if Ω is an open ball minus its center x , then u is radially symmetric with respect to x .(3) if Ω is a n -sided regular polygon, with n > , then u is invariant with respect to the dihedral group D n (of order n ).(4) if Ω is the product of rotationally invariant bounded domains, i.e., Ω = ω × ... × ω m , where ω j is abounded rotationnaly invariant domain of R n j , with n j > and N = n + ... + n m , then u inheritsthe same symmetry, i.e., u ( x ) = v ( | x | , ..., | x m | ) a.e. in Ω . This case was already addressed underthe additional assumption that u is smooth in Remark 2.1 in [6].(5) if Ω is a cylinder with ρ -invariant cross section, i.e., Ω = ω × U, where ω is a ρ -invariant boundeddomain of R k , k N − and U is a domain of R N − k , then u ( x ) = u ( ρ ( x ) , x k +1 , ..., x N ) a.e.on Ω ( here x := ( x , ..., x k ) ∈ R k . ) (6) any bounded domain of "revolution". 2. Proofs
Proof of Theorem 5.
We distinguish two case : either f ′ ( t ) for any t > or there exists ¯ t > such that f ′ (¯ t ) > . Inthe first case, by convexity of f, we have that f ′ (0) f ′ ( t ) for any t > , then f is also globallyLipschitz-continuous on [0 , + ∞ ) . So f ( u ) ∈ L (Ω) and u ∈ C (Ω) by standard elliptic estimates (plusbootstrap and Sobolev imbedding). The claim follows by taking f k = f and u k = u for any integer k > .In the second case, the convexity of f implies the existence of t > ¯ t that f ( t ) , f ′ ( t ) > for any t > t .Set k := ⌊ t ⌋ + 1 (here by ⌊ t ⌋ we denote the integer part of t ) and, for any integer k > k and t > ,we set(14) f k ( t ) := ( f ( t ) if t k,f ( k ) + f ′ ( k )( t − k ) if t > k.
6. Here H c (Ω) denotes the subset of H (Ω) whose members take the constant value c > on ∂ Ω . That is, H c (Ω) = { u ∈ H (Ω) : u − c ∈ H (Ω) } . In particular, for c = 0 , that set boils down to H (Ω) .7. An open ball, an open ball minus its center or an annulus.8. Here, for any j ∈ { , ..., m } , x j denotes a generic point of ω j ⊂ R n j in such a way that x := ( x , ..., x m ) ∈ R N . LOUIS DUPAIGNE AND ALBERTO FARINA
Clearly, f k is a convex function of class C ([0 , + ∞ )) ∩ C , ([0 , + ∞ )) and f k ր f pointwise in [0; + ∞ ) (recall that f ( t ) , f ′ ( t ) > for any t > t ). Moreover we have(15) f ′ k ( t ) f ′ ( t ) ∀ k > k , ∀ t > and(16) f k ( t ) > min t ∈ [0 ,k ] f ( t ) := c o ( f ) , ∀ k > k , ∀ t > . In particular, if f is nonnegative, then any function f k is nonnegative too.Since any f k is globally Lipschitz-continuous on [0 , + ∞ ) , we can use the (standard) method of sub andsupersolution in H to obtain a stable weak solution to (6) satisfying (7). To this end we observe that u ∈ H (Ω) is a nonnegative weak supersolution to(17) ( − ∆ v k = f k ( v k ) in Ω ,v k − u ∈ H (Ω) , since f k f on [0 , + ∞ ) implies that f k ( u ) ϕ f ( u ) ϕ a.e. in Ω , for any nonnegative ϕ ∈ C ∞ c (Ω) . So(18) ˆ Ω f k ( u ) ϕ ˆ Ω f ( u ) ϕ = ˆ Ω ∇ u ∇ ϕ ∀ ϕ ∈ C ∞ c (Ω) , ϕ > in Ω by (1) and then the above inequality holds true for any nonnegative ϕ ∈ H (Ω) by a standard densityargument. Also, is a weak subsolution to (17), since f (0) > by assumption and (0 − u ) + ≡ ∈ H (Ω) .Since u a.e. in Ω , the method of sub and supersolution in H provides a weak solution v k to (17) suchthat v k u a.e. in Ω and which is minimal in the following sense : given any weak supersolution u ∈ H of (17) such that u u a.e. in Ω , we have that v k u a.e. in Ω . From the latter property weimmediately infer that v k v k +1 u a.e. in Ω . Also, by standard elliptic estimates we have v k ∈ C ,αloc (Ω) for any α ∈ (0 , . By the convexity of f k and (15) we see that f ′ k ( v k ) f ′ k ( u ) f ′ ( u ) a.e. in Ω , hence(19) ˆ Ω f ′ k ( v k ) ϕ ˆ Ω f ′ ( u ) ϕ ˆ Ω |∇ ϕ | ∀ ϕ ∈ C c (Ω) and so v k is a stable weak solution to (6).To prove (7) we test (17) with u − v k ∈ H (Ω) to obtain(20) ˆ Ω ∇ v k ∇ ( u − v k ) = ˆ Ω f k ( v k )( u − v k ) and so(21) ˆ Ω |∇ v k | = ˆ Ω ∇ v k ∇ u − ˆ Ω f k ( v k )( u − v k ) ˆ Ω ∇ v k ∇ u − c ( f ) ˆ Ω ( u − v k ) thanks to (16) and u − v k > a.e. in Ω . Then ˆ Ω |∇ v k | ˆ Ω |∇ v k | + 12 ˆ Ω |∇ u | + | c ( f ) | ˆ Ω u hence(22) k∇ v k k L (Ω) k∇ u k L (Ω) + 2 | c ( f ) |k u k L (Ω) . We also have k v k k L (Ω) k u k L (Ω) since v k u a.e. in Ω . Then ( v k ) is bounded in H (Ω) andtherefore we may and do suppose that (up to subsequences) v k ⇀ v in H (Ω) , v k −→ v in L (Ω) , v k ր v a.e. on Ω for some v ∈ H (Ω) , as k → ∞ (actually the whole original sequence converges a.e. to v, sincewe already know that it is monotone nondecreasing). In particular we have v u a.e. in Ω . Also notethat v − u ∈ H (Ω) , since v k − u ⇀ v − u in H (Ω) .To prove that v k −→ v in H (Ω) we test (17) with v − v k ∈ H (Ω) to get(23) ˆ Ω ∇ v k ∇ ( v − v k ) = ˆ Ω f k ( v k )( v − v k )
9. Just consider the solution obtained by the standard monotone iterations procedure starting from the subsolution u ≡ . EGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS. 7 and so(24) ˆ Ω |∇ v k | = ˆ Ω ∇ v k ∇ v − ˆ Ω f k ( v k )( v − v k ) ˆ Ω ∇ v k ∇ v + ˆ Ω | c ( f ) | ( v − v k ) thanks to (16) and v − v k > a.e. in Ω . Recalling that v k v a.e. on Ω we then have(25) ˆ Ω | v k | + |∇ v k | ˆ Ω | v | + ˆ Ω ∇ v k ∇ v + ˆ Ω | c ( f ) | ( v − v k ) and so(26) lim sup k v k k H k v k H since v k ⇀ v in H (Ω) and v k → v in L (Ω) . Inequality (26) and v k ⇀ v in H (Ω) imply the strongconvergence in H (Ω) .To proceed further we note that f ( v ) ∈ L loc (Ω) since(27) c ( f ) f k ( v ) f ( v ) = f ( v ) { v k } + f ( v ) { v>k } sup t ∈ [0 ,k ] f ( t ) + f ( u ) , where in the latter we have used that f ′ ( t ) > for t > k and v ≤ u a.e in Ω .Next we prove that − ∆ v = f ( v ) in D ′ (Ω) . This follows by passing to the limit in (17) by the Lebesgue’sdominated convergence theorem, after having observed that f k ( v k ) −→ f ( v ) a.e. in Ω , and c ( f ) f k ( v k ) f ( v k ) = f ( v k ) { v k k } + f ( v k ) { v k >k } sup t ∈ [0 ,k ] f ( t ) + f ( u ) holds true.So far we have proved that(28) − ∆ v = f ( v ) in D ′ (Ω) v − u ∈ H (Ω) , v u a.e. on Ω , and therefore from Theorem 3 we deduce that either u = v in Ω or f ( t ) = a + λ t for all t ∈ (inf Ω v, sup Ω u ) and some a ∈ R , u, v ∈ C ∞ (Ω) and u − v is a positive first eigenfunction of − ∆ with homogeneous Dirichletboundary conditions. In the first case we are done, while in the second one we distinguish two subcases :either sup u = + ∞ or not. In the first subcase f is necessarily globally Lipschitz-continuous on [0 , + ∞ ) and the conclusion follows by taking f k = f and u k = u for every k . If sup u < + ∞ , the conclusionfollows by taking the sequences ( f k ) k > ¯ k and ( u ) k > ¯ k , where ¯ k is any integer satisfying ¯ k > sup u . Proof of Theorem 7.
When N = 1 , any u ∈ H loc (Ω) is continuous by Sobolev imbedding and so is f ( u ) . This implies u ∈ C and f ( u ) ∈ C , so u is of class C by using equation (1). So assume that N ≥ and that u is locallystable. For any point x ∈ Ω pick a ball B ( x , r ) ⊂ Ω in which u is stable and set B = B ( x , r ) .By Theorem 5 there is a nondecreasing sequence ( f k ) of functions in C ([0 , + ∞ )) ∩ C , ([0 , + ∞ )) suchthat f k ր f pointwise in [0; + ∞ ) and a nondecreasing sequence ( u k ) of functions in H (Ω) ∩ C (Ω) suchthat u k is a stable weak solution to (6) such that (7) holds.Since N an application of Theorem A to u k yields k u k k C β ( B ( x , r )) C k u k k L ( B ) C | B | k u k k L ( B ) C ′ k u k L ( B ) where β ∈ (0 , depends only on N , while C > depends on N and r . In particular, the sequence ( u k ) is bounded in C β ( B ( x , r )) and the Ascoli-Arzelà’s theorem then implies that a subsequence mustconverge uniformly to some continuous function v on B ( x , r )) . This function must coincide with u on B ( x , r )) , since we already know that u k −→ u a.e. on Ω . The boundedness of u on B ( x , r ) andstandard elliptic theory imply u ∈ C ,αloc ( B ( x , r )) for every α ∈ (0 , . Since x is an arbitrary point of Ω , this concludes the proof. (cid:3) Proof of Theorem 10.
1) In is enough to treat the case u . We first prove that u is radially symmetric. For ρ ∈ O ( N ) we set u ρ ( x ) := u ( ρx ) , x ∈ B . Then u ρ ∈ H ( B ) is a stable solution to (10) since u is so. We can therefore apply
10. That is a function u k satisfying ´ Ω ∇ u k ∇ ϕ = ´ Ω f k ( u k ) ϕ , for all ϕ ∈ H (Ω) LOUIS DUPAIGNE AND ALBERTO FARINA
Theorem 17 below to get that u and u ρ are ordered solutions. If u u ρ , an application of Theorem 3 wouldgive u, u ρ ∈ C ∞ and, either u < u ρ or u > u ρ in B . The latter are clearly impossible, since u (0) = u ρ (0) .Thus, u ≡ u ρ for any ρ ∈ O ( N ) , and so u is radially symmetric in B . Since u is a radially symmetricmember of H ( B ) , we have that u ∈ C ( B \ { } ) and by standard elliptic regularity, u ∈ C ,αloc (( B \ { } ) ,for any α ∈ (0 , . Hence we can write u ( x ) = v ( r ) , r = | x | ∈ (0 , R ] , and so v ∈ C ((0 , R ]) is a classicalsolution to the ode − ( r N − v ′ ) ′ = r N − f ( v ) in (0 , R ] . The latter clearly implies v ∈ C ((0 , R ]) but alsothat (29) v ′ < in ( R − ǫ, R ) , for some ǫ ∈ (0 , R ) . Indeed, if f (0) > the Hopf’s lemma yields v ′ ( R ) < (recall that we are supposingthat u and so (29) follows. When f (0) < we have ( r N − v ′ ) ′ = − r N − f ( v ) > in an interval ofthe form ( R − ǫ, R ) , thus r → r N − v ′ is strictly increasing in ( R − ǫ, R ) . The latter and the fact that v ′ ( R ) (recall that v > in (0 , R ) and v ( R ) = 0 ) imply (29).To conclude it is enough to prove that v ′ < on (0 , R ) (this also implies that u > in B \ { } ).Suppose not, then(30) r := inf { r ∈ (0 , R ) : v ′ < on ( r, R ) } . is well-defined, r belongs to (0 , R ) and v ′ ( r ) = 0 . We have two cases : either there exists z ∈ (0 , r ) suchthat v ′ ( z ) = 0 , or v ′ has a sign on (0 , r ) (i.e., either v ′ < or v ′ > on (0 , r ) ). Let us show that bothof them are impossible. In the first case we observe that w := u r , the radial derivative of u , is of class C ( A z,r ) and satisfy(31) − ∆ w + ( N − wr = f ′ ( u ) w on A z,r , where A z,r := { x ∈ R N : z < | x | < r } . We can then multiply (31) by w , integrate by parts and find(32) ˆ A z,r |∇ w | − ˆ A z,r f ′ ( u ) w = − ˆ A z,r ( N − w r . On the other hand w ∈ C ( A z,r ) and u is stable on A z,r , so w can be used as test function in thestability condition satisfied by u to obtain ˆ A z,r |∇ w | − ˆ A z,r f ′ ( u ) w > . The latter and (32) give w = 0 on A z,r and so w in the open annulus A z,R . Since w solves thelinear equation − ∆ w + ( ( N − r − f ′ ( u )) w = 0 on A z,R , the strong maximum principle implies w ≡ on A z,R , and so v ′ ≡ on ( z, R ) , which contradicts (29). In the second case, if v ′ < on (0 , r ) then w on A ,R and, as before, w ≡ on A ,R by the the strong maximum principle. The latter is againin contradiction with (29). Hence we are left with v ′ > on (0 , r ) . In this case, the latter and thedefinition of r imply u v ( r ) on B \ { } , so u ∈ L ∞ ( B ) and then u ∈ C ( B ) by standard ellipticestimates. To achieve a contradiction, we first observe that − ∆ w + ( N − wr = f ′ ( u ) w on the annulus A r,r := { x ∈ R N : r < | x | < r } , for every r ∈ (0 , r ) , we then multiply the equation by w , integrate byparts to get(33) ˆ A r,r |∇ w | − ˆ A r,r f ′ ( u ) w + ˆ A r,r ( N − w r = ˆ ∂A r,r ∂w∂ν w. Note that ´ ∂A r,r ∂w∂ν w −→ , as r → + . Indeed, the function(34) ˜ w ( x ) := ( w ( x ) if x ∈ B \ { } , if x = 0 ,
11. In view of (29) we could have used the moving planes procedure to get the strict monotonicity of u in the radialdirection. However, we have chosen to give an elementary proof of this fact, which highlights the role played by the stabilityassumption on u . EGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS. 9 is of class C , ( B ) and therefore | ´ ∂A r,r ∂w∂ν w | ´ S r |∇ ˜ w || w | k∇ ˜ w k L ∞ ( B ) σ ( S r ) v ′ ( r ) −→ (here S r is the sphere { x ∈ R N : | x | = r } and σ ( S r ) denotes its measure). Also observe that ˆ A r,r |∇ w | − ˆ A r,r f ′ ( u ) w + ˆ A r,r ( N − w r −→ ˆ B (0 ,r ) |∇ ˜ w | − ˆ B (0 ,r ) f ′ ( u ) ˜ w + ˆ B (0 ,r ) ( N −
1) ˜ w r by monotone and dominated convergence theorems. By gathering together all those information we areled to(35) ˆ B (0 ,r ) |∇ ˜ w | − ˆ B (0 ,r ) f ′ ( u ) ˜ w = − ˆ B (0 ,r ) ( N −
1) ˜ w r . Finally, since u is stable on B , then u is stable on B (0 , r ) too. Thanks to item i) of Proposition 15) wecan use w ∈ C , ( B (0 , r )) ⊂ H ( B (0 , r )) as test function in the stability condition satisfied by u to get ´ B (0 ,r ) |∇ ˜ w | − ´ B (0 ,r ) f ′ ( u ) ˜ w > . As before, the latter and (35) imply that w ≡ on B (0 , r ) \ { } ,which is impossible since we are assuming that v ′ > in (0 , r ) . If N and f > , then u is C in a neighborhood of the origin by item (1) of Theorem 7. Thiscompletes the proof of item (1).2) By combining item (1) and item (2) of Theorem 7 we have that u ∈ C ( B ) . Then, either u ≡ or u > in B by the strong maximum principle. The desired conclusion then follows by a celebrated resultof Gidas, Ni and Nirenberg.3) Let H be any hyperplane through the origin and let ρ = ρ ( H ) ∈ O ( N ) be the correspondingreflection with respect to H . Set u ρ ( x ) := u ( ρx ) , x ∈ Ω . Then u ρ ∈ H (Ω) is a stable solution to (12)since u is so. By Theorem 17, u and u ρ are ordered solutions. If u u ρ , an application of Theorem 3would give u, u ρ ∈ C ∞ and, either u < u ρ or u > u ρ in Ω . The latter are clearly impossible, since u = u ρ on the hyperplane H . Thus, u ≡ u ρ and so u is radially symmetric in Ω , since H is arbitrary. Since u is a radially symmetric member of H (Ω) , we have that u ∈ C (Ω) and standard elliptic theory imply u ∈ C ,αloc (Ω) , for any α ∈ (0 , . The required regularity then follows, as in item (1), by analysing the odesatisfied by the radial profile of u . (cid:3) Some auxiliary results
In this section we prove some results for stable solutions in H . These results have been used in theproofs of our main results and some of them are of independent interest. In what follows, when Ω isbounded, we shall denote by λ = λ (Ω) > the principal eigenvalue of − ∆ with homogeneous Dirichletboundary conditions. Lemma 14.
Let Ω be a bounded domain of R N , N > and let f ∈ C ( R ) such that f ( t ) > λ t for t > .Assume that u ∈ H (Ω) and f ( u ) ∈ L loc (Ω) .If u solves (36) ( − ∆ u > f ( u ) in D ′ (Ω) u > a.e. on Ω , then u ≡ , f (0) = 0 and u is a solution to (37) − ∆ u = f ( u ) in D ′ (Ω) which is not stable.
12. Since u ∈ C ( B ) is a radial function we have ∇ u (0) = 0 , w := u r ∈ C ( B \ { } ) and | w ( x ) | ≤ C | x | in B , for someconstant C > . Then ˜ w ∈ C ( B ) , | v ′ ( r ) | Cr in [0 , R ] and so | v ′′ ( r ) | C in (0 , R ] , for some constant C > , byusing the ode satisfied by v . Thus, for any x, y ∈ B such that | y | > | x | > we have : | w ( y ) − w ( x ) | = | v ′ ( | y | ) − v ′ ( | x | ) | ´ | y || x | | v ′′ ( t ) | dt sup ξ ∈ [ | x | , | y | ] | v ′′ ( ξ ) | ( | y | − | x | ) sup r ∈ B \{ } | v ′′ ( r ) | y − x | C | y − x | . Then, ˜ w ∈ C , ( B ) . Proof.
By the strong maximum principle (for superharmonic functions in H ) either u ≡ or u > a.e. on Ω . Let us prove that the latter is impossible. To this end, let us suppose that u > and let h ∈ H (Ω) be the unique weak solution to(38) ( − ∆ h = 0 in Ω , u − h ∈ H (Ω) . Then h u a.e. on Ω by the maximum principle, and h < u a.e. on Ω by the strong maximumprinciple. Indeed, h = u would imply − ∆( u − h ) = f ( u ) > λ u > a.e. on Ω , a contradiction.Let φ ∈ H (Ω) be a positive function associated to the eigenvalue λ . Since f ( u ) > a.e. on Ω , astandard density argument and Fatou’s Lemma imply ˆ Ω ∇ u ∇ ϕ > ˆ Ω f ( u ) ϕ ∀ ϕ ∈ H (Ω) , ϕ > a.e. in Ω , and therefore we can use φ in the latter to get ˆ Ω ∇ u ∇ φ > ˆ Ω f ( u ) φ . Then, ˆ Ω λ uφ < ˆ Ω f ( u ) φ ˆ Ω ∇ u ∇ φ = ˆ Ω ∇ ( u − h ) ∇ φ + ˆ Ω ∇ h ∇ φ = ˆ Ω ∇ ( u − h ) ∇ φ + 0 where we have used the property of f , as well as the fact that h solves (38). In addition, by definition of φ and u − h ∈ H (Ω) , we also have ´ Ω ∇ ( u − h ) ∇ φ = ´ Ω λ ( u − h ) φ , which leads to ˆ Ω λ uφ < ˆ Ω f ( u ) φ ˆ Ω ∇ u ∇ φ = ˆ Ω ∇ ( u − h ) ∇ φ = ˆ Ω λ ( u − h ) φ ˆ Ω λ uφ a contradiction. Therefore, u ≡ and f (0) − ∆ u = 0 . The latter and the assumption on f give f (0) = 0 and so u solves (37). Since f ′ (0) > λ = inf { ´ Ω |∇ u | ´ Ω u : u ∈ H (Ω) , u } we immediately see that u = 0 cannot be a stable solution to (37). (cid:3) Proposition 15.
Let Ω be an open set of R N , N > , and let f ∈ C ([0; + ∞ )) be a convex function.i) Let u ∈ L loc (Ω) such that u > a.e. in Ω and (39) f ′ ( u ) ∈ L loc (Ω) , ˆ Ω f ′ ( u ) φ ˆ Ω |∇ φ | ∀ φ ∈ C ∞ c (Ω) . Then (40) f ′ ( u ) ϕ ∈ L (Ω) , ˆ Ω f ′ ( u ) ϕ ˆ Ω |∇ ϕ | ∀ ϕ ∈ H (Ω) . ii) Let u, v ∈ H (Ω) such that f ( u ) , f ( v ) ∈ L loc (Ω) , ( u − v ) + ∈ H (Ω) and v, u a.e. in Ω . Alsoassume that u satisfies (39) . Then, ( f ( u ) − f ( v ))( u − v ) + ∈ L (Ω) . iii) Let u, v ∈ L loc (Ω) such that f ( u ) , f ( v ) ∈ L loc (Ω) and v u a.e. in Ω .Then, f ′ ( v )( u − v ) ∈ L loc (Ω) . Proof. i) By convexity of f and (39) we have(41) f ′ (0) f ′ ( u ) a.e. in Ω and(42) ˆ Ω [ f ′ ( u )] + φ ˆ Ω |∇ φ | + ˆ Ω [ f ′ ( u )] − φ ∀ φ ∈ C ∞ c (Ω) . From (41) we deduce that [ f ′ ( u )] − [ f ′ (0)] − and so [ f ′ ( u )] − ∈ L ∞ (Ω) and [ f ′ ( u )] − ϕ ∈ L (Ω) for any ϕ ∈ H (Ω) . Now, for any ϕ ∈ H (Ω) , let ( φ n ) be a sequence of functions in C ∞ c (Ω) such that φ n −→ ϕ in H (Ω) and a.e. in Ω . Using φ = φ n in (42) and Fatou’s Lemma we immediately get that (42) holdstrue for any ϕ ∈ H (Ω) . In particular, [ f ′ ( u )] + ϕ ∈ L (Ω) and the two claims of (40) follow. EGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS. 11 ii) The convexity of f and ( u − v ) + > a.e. on Ω imply(43) f ′ (0)[( u − v ) + ] f ′ ( v )[( u − v ) + ] ( f ( u ) − f ( v ))( u − v ) + f ′ ( u )[( u − v ) + ] a.e. on Ω and so(44) ( f ( u ) − f ( v ))( u − v ) + ∈ L (Ω) thanks to ( u − v ) + ∈ H (Ω) and item i).iii) By convexity of f we have f ( u ) − f ( v ) > f ′ ( v )( u − v ) > f ′ (0)( u − v ) which implies the desired conclusion. (cid:3) Theorem 16.
Let Ω be a bounded domain of R N , N > and let f ∈ C ([0; + ∞ )) be a convex function.Assume that u, v ∈ H (Ω) satisfy u − v ∈ H (Ω) , v u a.e. on Ω , f ( u ) , f ( v ) ∈ L loc (Ω) and both u and v are solution to (45) − ∆ w = f ( w ) in D ′ (Ω) . If f ′ ( u ) ∈ L loc (Ω) and u is stable, then either u ≡ v or f ( t ) = a + λ t for all t ∈ (inf Ω v, sup Ω u ) andsome a ∈ R , u, v ∈ C ∞ (Ω) and u − v is a positive first eigenfunction of − ∆ with homogeneous Dirichletboundary conditions. Proof.
By item ii) of Proposition 15 we have(46) ( f ( u ) − f ( v ))( u − v ) ∈ L (Ω) , note that u − v = ( u − v ) + since v u a.e. in Ω .We claim that(47) ˆ Ω |∇ ( u − v ) | = ˆ Ω ( f ( u ) − f ( v ))( u − v ) . To this end, recall that(48) ˆ Ω ∇ ( u − v ) ∇ ϕ = ˆ Ω ( f ( u ) − f ( v )) ϕ ∀ ϕ ∈ C ∞ c (Ω) by assumption. Then, since f ( u ) − f ( v ) ∈ L loc (Ω) , a standard approximation argument and Lebesgue’sdominated convergence theorem yield that (48) holds true for any ϕ ∈ H (Ω) ∩ L ∞ (Ω) with compactsupport. Now we prove that (48) holds true for any ϕ ∈ H (Ω) with compact support and such that ϕ u − v a.e. on Ω . Indeed, we can apply (48) with ϕ n = T n ( ϕ ) , where T n = T n ( t ) denotes thetruncation function at level n > , and write(49) ˆ Ω ∇ ( u − v ) ∇ ϕ n = ˆ Ω ( f ( u ) − f ( v )) ϕ n , ∀ n > . Since ϕ n → ϕ in H (Ω) and a.e. on Ω , and (46) is in force, an application of Lebesgue’s dominatedconvergence theorem we get(50) ˆ Ω ∇ ( u − v ) ∇ ϕ = ˆ Ω ( f ( u ) − f ( v )) ϕ ∀ ϕ ∈ H (Ω) , supp ( ϕ ) ⊂⊂ Ω , ϕ u − v a.e. in Ω . To conclude the proof of (47) we proceed as follows. Let ( ρ n ) be a sequence of nonnegative functions in C ∞ c (Ω) such that ρ n → u − v in H (Ω) and a.e. on Ω and set w n := min { u − v, ρ n } . Then, w n ∈ H (Ω) , supp ( w n ) ⊂ supp ( ρ n ) ⊂⊂ Ω and w n u − v a.e. on Ω . We can therefore apply (50) with ϕ = w n toget the desired conclusion (47), since w n → u − v in H (Ω) and a.e. on Ω , and (46) is in force.
13. Actually the real number a is unique and its value is given by − λ ´ Ω φ h ´ Ω φ , where φ is a positive first eigenfunctionof − ∆ with homogeneous Dirichlet boundary conditions and h ∈ H (Ω) is the unique weak solution of − ∆ h = 0 in Ω with u − h ∈ H (Ω) (to see this, use φ as test function in the weak formulation of − ∆ u = a + λ u and the fact that h isharmonic) and also note that h is nonnegative by the maximum principle. In particular, u, v ∈ H (Ω) ⇐⇒ a = 0 . Also notethat, for every a there exist solutions u, v for which the second alternative of the theorem occurs. Indeed, the functions u t := − aλ + tφ , t > are suitable. By convexity of f we have f ′ (0) f ′ ( v ) f ′ ( u ) a.e. on Ω , hence f ′ ( v ) ∈ L loc (Ω) and [ f ′ ( v )] − [ f ′ (0)] − a.e. in Ω . Also observe that − ∆( u − v ) = f ( u ) − f ( v ) > f ′ ( v )( u − v ) in D ′ (Ω) and so also − ∆( u − v ) + [ f ′ ( v )] − ( u − v ) > [ f ′ ( v )] + ( u − v ) > in D ′ (Ω) , since u − v ∈ H (Ω) is nonnegative a.e. in Ω . The strong maximum principle (see e.g. Theorem 8.19 in [14]) yields either u ≡ v, and we are done, or u > v a.e. on Ω . In the remaining part of the proof we assume that the latter possibility is in force andwe set I := inf Ω v, S := sup Ω u .Now we combine (47) with (40) with ϕ = u − v to get ˆ Ω (cid:0) f ( u ) − f ( v ) − f ′ ( u )( u − v ) (cid:1) ( u − v ) > . By convexity of f , the integrand in the above inequality is nonpositive and so (cid:0) f ( u ) − f ( v ) − f ′ ( u )( u − v ) (cid:1) ( u − v ) = 0 a.e. in Ω and then f ( u ) − f ( v ) − f ′ ( u )( u − v ) = 0 a.e. in Ω since u > v a.e. on Ω . The latter implies that, for almost every x ∈ Ω , the function f must be affine onthe open interval ( v ( x ) , u ( x )) . Now we prove that f is an affine function on the the interval ( I, S ) . To thisend we first consider the case in which one of the two solutions is a constant function. If v is constant, say v ≡ c , then I = c and for almost every x ∈ Ω the function f is affine on the open interval ( I, u ( x )) . Sinceany two intervals of this form intersect and f is affine on each of them, we see that f must be the sameaffine function on both intervals. This implies that f is an affine function over the entire interval ( I, S ) , as claimed. The same argument applies if u is constant. It remains to consider the case where neither v nor u are constant. Since v is not constant we can find x ∈ Ω such that(51) v ( x ) < u ( x ) and f ( t ) = a + bt for all t ∈ ( v ( x ) , u ( x )) , |{ v > v ( x ) }| > , |{ v v ( x ) }| > , where we denoted by | X | the Lebesgue measure of any measurable set X ⊂ Ω .Let ( α, β ) be the largest open interval containing ( v ( x ) , u ( x )) and such that f ( t ) = a + bt for all t ∈ ( α, β ) . To conclude the proof it is enough to show that ( I, S ) ⊂ ( α, β ) . Let us first prove that I > α .Assume to the contrary that I < α, then the latter and (51) give
I < α v ( x ) < sup Ω v and so we canfind an integer m > such that(52) I inf ω m v < α < sup ω m v where { ω m , m > } is a countable family of open connected sets of R N such ω m ⊂ ω m +1 ⊂⊂ Ω and Ω = ∪ m > ω m (such a family of open connected sets exists thanks to the fact that Ω is open, bounded andconnected). Recall that u − v ∈ H (Ω) is positive a.e on Ω and satisfies − ∆( u − v ) + [ f ′ ( v )] − ( u − v ) > [ f ′ ( v )] + ( u − v ) > in D ′ (Ω) , where [ f ′ ( v )] − ∈ L ∞ (Ω) , then the strong maximum principle yields(53) u − v > c ( m ) > a.e. on ω m .Now, if we set ǫ := min { c ( m )2 , α − inf ω m v } > , we have I inf ω m v < α − ǫ < α < sup ω m v and so(54) |{ v α − ǫ } ∩ ω m | > , |{ v > α − ǫ } ∩ ω m | > . Since ω m is open and connected we can use the "intermediate value theorem" for functions in a Sobolevspace (see e.g. Théorème 1 of [8]) to get |{ α − ǫ v α − ǫ } } ∩ ω m | > . This fact and (53) prove theexistence of x ǫ ∈ ω m such that α − ǫ v ( x ǫ ) α − ǫ and u ( x ǫ ) − v ( x ǫ ) > c ( m ) , whose combination leadsto u ( x ǫ ) > α + c ( m )2 and v ( x ǫ ) < α. Therefore, f must satisfy f ( t ) = a + bt for all t ∈ ( v ( x ǫ ) , β ) , whichcontradicts the definition of ( α, β ) . This proves that I > α . Now we show that S β . If β = + ∞ we aredone, so we suppose that β ∈ R . In this case we observe that inf Ω u < β since otherwise we would have u > β a.e. on Ω and so the strong maximum principle (applied to u − β ) would give u > β a.e. on Ω (recall that v is not constant by assumption). Hence there would exist ¯ x ∈ Ω such that v (¯ x ) < β < u (¯ x ) .This would imply that f ( t ) = a + bt for all t ∈ ( α, u (¯ x )) , contradicting the definition of ( α, β ) . If S > β
EGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS. 13 (and since inf Ω u < β ) the same argument used to prove α I can be applied to the solution u to geta contradiction (again that f would satisfy f ( t ) = a + bt for all t ∈ ( α, u ( y )) , for some y ∈ Ω such that u ( y ) > β ). Therefore S β and f ( t ) = a + bt for all t ∈ ( I, S ) . Hence, u and v are of class C (Ω) bystandard elliptic estimates. Finally, u − v ∈ H (Ω) solves − ∆( u − v ) = f ( u ) − f ( v ) = b ( u − v ) in Ω and u − v > in Ω . Thererefore u − v is a positive eigenfunction and so we necessarily have b = λ . Thisconcludes the proof. (cid:3) Theorem 17.
Let Ω be a bounded domain of R N , N > and let f ∈ C ([0; + ∞ )) be a convex function.Assume that u, v ∈ H (Ω) satisfy u − v ∈ H (Ω) , f ( u ) , f ( v ) ∈ L loc (Ω) and both u and v are solution to (55) ( − ∆ w = f ( w ) in D ′ (Ω) w > a.e. on Ω , If u satisfies f ′ ( u ) ∈ L loc (Ω) and u is stable, then u and v are ordered, namely, one of the following threecases holds true : u < v a.e. in Ω , u ≡ v a.e. in Ω or u > v a.e. in Ω . Proof.
By item ii) of Proposition 15 we have(56) ( f ( u ) − f ( v ))( u − v ) + ∈ L (Ω) In view of the above results we can follow the proof of Theorem 3 to obtain(57) ˆ Ω |∇ ( u − v ) + | = ˆ Ω ( f ( u ) − f ( v ))( u − v ) + . Using (40) with ϕ = ( u − v ) + , (57) and (43) we have ˆ Ω f ′ ( u )[( u − v ) + ] ˆ Ω |∇ ( u − v ) + | = ˆ Ω ( f ( u ) − f ( v ))( u − v ) + ˆ Ω f ′ ( u )[( u − v ) + ] and so ˆ Ω |∇ ( u − v ) + | − ˆ Ω f ′ ( u )[( u − v ) + ] = 0 . The latter and (40) imply that ( u − v ) + minimizes the the functional ψ −→ ´ Ω |∇ ψ | − ´ Ω f ′ ( u ) ψ over H (Ω) and therefore ( u − v ) + solves(58) − ∆( u − v ) + = f ′ ( u )( u − v ) + in D ′ (Ω) . Then(59) − ∆( u − v ) + + [ f ′ ( u )] − ( u − v ) + = [ f ′ ( u )] + ( u − v ) + > in D ′ (Ω) with [ f ′ ( u )] − ∈ L ∞ (Ω) , since f ′ ( u ) > f ′ (0) by convexity of f . By the strong maximum principle, either ( u − v ) + > a.e. in Ω or ( u − v ) + = 0 a.e. in Ω . That is, either u > v a.e. in Ω or u v a.e. in Ω . In the lattercase we have − ∆( v − u ) = f ( v ) − f ( u ) > f ′ ( u )( v − u ) in D ′ (Ω) and so also − ∆( v − u ) + [ f ′ ( u )] − ( v − u ) > [ f ′ ( u )] + ( v − u ) > in D ′ (Ω) , since v − u ∈ H (Ω) is nonnegative a.e. in Ω . As above, another applicationof the strong maximum principle yields either u ≡ v or v > u a.e. on Ω . (cid:3) By combining Lemma 14 and Theorem 3 we immediately obtain the following classification result forstable solutions in H (Ω) . Theorem 18.
Let Ω be a bounded domain of R N , N > and let f ∈ C ( R ) be a convex function suchthat f (0) = 0 . Assume that u ∈ H (Ω) , f ( u ) ∈ L loc (Ω) and that u is a stable solution to (60) ( − ∆ u = f ( u ) in D ′ (Ω) u > a.e. on Ω . Then, either u ≡ or f ( t ) = λ t on (0 , sup Ω u ) and u ∈ C ∞ (Ω) ∩ H (Ω) is a positive first eigenfunctionof − ∆ with homogeneous Dirichlet boundary conditions.Remark . If u ≡ , then necessarily f ′ (0) λ by Lemma 14. Also observe that for any α λ thereis a convex function f satisfying f (0) = 0 , f ′ (0) = α and such that u ≡ is a stable solution to (3). Anexample is provided by f ( u ) = u + αu . Proof. v ≡ is a solution to (3) since f (0) = 0 . Then, an application of Theorem 3 provides thedesired results. Indeed, since u ∈ H (Ω) , we have a = 0 (as observed in the footnote to Theorem 3).Therefore f ( t ) = λ t on (inf Ω v, sup Ω u ) = (0 , sup Ω u ) . (cid:3) Proposition 20.
Let Ω be a bounded domain of R N , N > and let f ∈ C ([0; + ∞ )) be a convexfunction. Assume that u, v ∈ H loc (Ω) satisfy v < u a.e. on Ω , f ( u ) , f ( v ) ∈ L loc (Ω) and (61) ( − ∆ u > f ( u ) in D ′ (Ω) , − ∆ v f ( v ) in D ′ (Ω) . Then (62) f ′ ( v ) ∈ L loc (Ω) , ˆ Ω f ′ ( v ) ϕ ˆ Ω |∇ ϕ | ∀ ϕ ∈ C ∞ c (Ω) . In particular, if v is a solution to − ∆ v = f ( v ) in D ′ (Ω) , then v is stable. Proof.
Recall that, by convexity of f , we have f ( u ) − f ( v ) > f ′ ( v )( u − v ) and [ f ′ ( v )] − [ f ′ (0)] − a.e.in Ω . Therefore, [ f ′ ( v )] − ∈ L ∞ (Ω) and, using (61), we obtain − ∆( u − v ) + [ f ′ ( v )] − ( u − v ) > f ( u ) − f ( v ) + [ f ′ ( v )] − ( u − v ) > [ f ′ ( v )] + ( u − v ) > in D ′ (Ω) . By the strong maximum principle and u − v > a.e. in Ω we then get(63) ∀ ω ⊂⊂ Ω u − v > c ( ω ) > a.e. on ω ,where c ( ω ) is a positive constant depending on the open subset ω . The latter implies that u − v ∈ L ∞ loc (Ω) and so f ′ ( v ) ∈ L loc (Ω) , thanks to item iii) of Proposition 15. This proves the first claim of (62). To provethe second one we recall that f ′ ( v )( u − v ) ∈ L loc (Ω) and use once again (61) to get(64) ˆ Ω ∇ ( u − v ) ∇ φ > ˆ Ω f ′ ( v )( u − v ) φ, ∀ φ ∈ C ∞ c (Ω) . As before, a standard approximation argument and Lebesgue’s dominated convergence theorem yieldthat (64) holds true for any φ ∈ H (Ω) ∩ L ∞ (Ω) with compact support. Therefore, for every ϕ ∈ C ∞ c (Ω) and recalling (63), we can then take φ = ϕ u − v in (64) and find(65) ˆ Ω ∇ ( u − v ) ∇ (cid:16) ϕ u − v (cid:17) > ˆ Ω f ′ ( v ) ϕ , ∀ ϕ ∈ C ∞ c (Ω) . Hence, ˆ Ω f ′ ( v ) ϕ ˆ Ω ϕ ∇ ( u − v ) u − v ∇ ϕ − ˆ Ω ϕ ( u − v ) |∇ ( u − v ) | , ∀ ϕ ∈ C ∞ c (Ω) . The second conclusion of (62) then follows by applying Young’s inequality to the first integral on ther-h-s of the latter inequality. The last claim is a consquence of (62). (cid:3)
Proof of Theorem 9. (1) Since u has finite Morse index, there exists a neighborhood of the boundary of the form Ω ǫ = { x ∈ Ω : dist ( x, ∂ Ω) < ǫ } such that u is stable in Ω ǫ (see point (5) below Definition 1). Thanks to Theorem5, it suffices to prove (9) in the case where u ∈ C ,α (Ω ǫ ) . Also, the estimate will follow if we prove thatfor some ρ ∈ (0 , ǫ ) and for every x ∈ Ω ρ , there exists a set I x such that | I x | ≥ γ and u ( x ) ≤ u ( y ) , for all y ∈ I x .To this end, we apply the moving-plane method. For y ∈ ∂ Ω , let n ( y ) denote the unit normal vectorto ∂ Ω , pointing outwards. Thanks to Lemmas 4.1 and 4.2 in [2], there exists a constant λ ∈ (0 , ǫ/ depending on Ω only, such that { x = y − tn ( y ) : 0 < t < λ , y ∈ ∂ Ω } ⊂ Ω In addition, in a fixed neighborhood of ∂ Ω , every point can be written in the form x = y − tn ( y ) , where < t < λ and y is the unique projection of x on ∂ Ω . Fix x ∈ ∂ Ω and n = n ( x ) . By applying thestandard moving-plane method in the cap Σ λ := { x ∈ Ω : 0 < − ( x − x ) · n < λ } , we deduce that(66) ∂ n u < in Σ λ , for every λ ∈ [0 , λ ] . EGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS. 15
Next, since Ω is uniformly convex, there exists a radius r > depending on Ω only, such that the geodesicball B = B ( n ( x ) , r ) ⊂ S N − can be realized as the set of normals at nearby points and so B ⊂ n ( ∂ Ω) .To see this, assume without loss of generality that x = 0 and that ∂ Ω coincides near x with the graphof some C function ϕ : R N − → R such that ϕ (0) = 0 , ∇ ϕ (0) = 0 and ∇ ϕ (0) is a diagonal matrix witheigenvalues bounded below by a positive constant (i.e. the directions of principal curvature of ∂ Ω at x coincide with the canonical basis of R N − ). Then, n ( x ) = (0 , . . . , , and for t = ( t , , . . . , ∈ R N − small, there holds n ( ϕ ( t )) = ( −∇ ϕ ( t ) , p |∇ ϕ ( t ) | = n ( x ) − t (cid:18) ∂ ϕ∂t (0) , , . . . , (cid:19) + o ( | t | ) and so n describes an arc of circle in the x direction as t varies in some small interval ( − r , r ) . Thisis also true (uniformly, since Ω is uniformly convex) in any direction e ∈ R N − and so a small geodesicball B = B ( n ( x ) , r ) ⊂ S N − can indeed be realized as the set of normals at nearby points. This in turnimplies that for all θ ∈ B , ∂ θ u < in Σ := (cid:26) x ∈ Ω : 14 λ < − ( x − x ) · n ( x ) < λ (cid:27) .Indeed, applying the moving-plane procedure at every point y ∈ ∂ Ω such that θ = n ( y ) , θ ∈ B , we have ∂ θ u < in { x ∈ Ω : 0 < − ( x − y ) · θ < λ } . By taking a smaller ball B if necessary, we may assume that | ( x − x ) · ( θ − n ( x )) + ( x − y ) · θ | < λ , for all x ∈ Σ and θ = n ( y ) ∈ B .Now, since − ( x − y ) · θ = − ( x − x ) · n ( x ) − ( x − x ) · ( θ − n ( x )) − ( x − y ) · θ , we have for any x ∈ Σ , λ = 14 λ + 34 λ > − ( x − y ) · θ > λ − λ = 0 and so, as claimed, for any x ∈ Σ , there holds ∂ θ u ( x ) < . Now take ρ = λ / . Fix a point x ∈ Ω ρ = { x ∈ Ω : dist ( x, ∂ Ω) < ρ } and let x denote its projection on ∂ Ω . On the one hand, u ( x ) ≤ u ( x ) , where x = x − ρn ( x ) . On the other hand, u ( x ) ≤ u ( z ) , for all z in the cone I x ⊂ Σ having vertex at x , opening angle B , and height λ / and the proof is complete.(2) Fix ǫ > λ > as above. According to Theorem 5, there exists a sequence of functions u k ∈ C ,α (Ω ǫ ) which are stable solutions of a semilinear elliptic equation in Ω ǫ and converge a.e. to u in Ω ǫ .If N ≤ , we may apply the interior estimate Theorem A to deduce that k u k k L ∞ (Ω λ \ Ω λ ) ≤ M k u k k L (Ω ǫ ) ≤ M k u k L (Ω) , for some constant M depending on Ω . Furthermore, if Ω is convex, we know that u k is monotone in thenormal direction i.e. (66) holds for u = u k , and so the inequality remains true all of Ω λ . Passing to thelimit k → + ∞ and using standard elliptic regularity, we deduce that u ∈ C ,α (Ω λ ) in this case. In thecase where f is nondecreasing on Ω is C , we can directly apply Theorem 5 combined to Theorem 1.5 in[5].(3) We write a generic point in R N as ( x, y ) ∈ R N − × R . Let B ′ be the unit ball in R N − , N − ≥ and Ω ⊂ R N the open set obtained by gluing the cylinder B ′ × ( − , to the unit half-ball centeredat ( x, y ) = (0 , − and to the unit half-ball centered at ( x, y ) = (0 , . Let λ n : [0 , → R + be a C increasing concave function such that λ n ( y ) = ny for y ∈ [0 , and λ n (2) = ( n + ρ n ) , where ρ n → .Extend λ n as an odd function on [ − , . Then, the domain Ω n = { ( x, λ n ( y )) : ( x, y ) ∈ Ω } is convexbut clearly not uniformly, nor even strictly. We let u n be the minimal solution to (1) with nonlinearity f ( u ) = 2( N − e u and domain Ω n . Since u = 0 and u = − | x | ) are ordered sub and supersolution tothe problem, u n is well-defined, stable and < u n < − | x | ) in Ω n . It already readily follows that the average | Ω n | k u n k L (Ω n ) of u n remains bounded. In addition, since − | x | ) is a strict supersolution of the equation, u n cannot be an extremal solution and so u n issmooth and strictly stable, i.e. its linearized operator has positive first eigenvalue.Recall that ρ n → and assume by contradiction that k u n k L ∞ ((Ω n ) ρn ) ≤ M for some constant M > .For ( x, y ) ∈ Ω , let v n ( x, y ) = u n ( x, λ n ( y )) . Then, v n = 0 on ∂ Ω and, letting µ n denote the inverse functionof λ n ,(67) − (∆ x + ( µ ′ n ) ∂ y + µ ′′ n ∂ y ) v n = f ( v n ) in Ω . We claim that k∇ v n k L ∞ (Ω) ≤ K, for some constant K > . To see this, we begin by estimating |∇ v n | on ∂ Ω . On the flat part of theboundary, we have a natural barrier : since v n < − | x | ) in Ω and v n = − | x | ) = 0 on ∂B ′ × ( − , ,we deduce that k∇ v n k L ∞ ( ∂B ′ × ( − , ≤ . The function ζ ( x, y ) = 1 − | ( x, y ) − (0 , | vanishes on theboundary of the half-ball centered at (0 , and satisfies − (∆ x + ( µ ′ n ) ∂ y + µ ′′ n ∂ y ) ζ = 2( N −
1) + 2( µ ′ n ) − µ ′′ n ∂ y ζ ≥ N − for y ≥ .Hence, a constant multiple of ζ can be used as a barrier on the half-ball centered at (0 , . Workingsimilarly with the other half-ball, we deduce that k∇ v n k L ∞ ( ∂ Ω) ≤ K on the whole boundary of Ω . Toextend the inequality to the whole of Ω , we observe that any partial derivative ∂ i v n solves the linearizedequation. Since v n is strictly stable (because this is the case for u n ), the linearized operator at v n haspositive first eigenvalue. It follows that k∇ v n k L ∞ (Ω) ≤ k∇ v n k L ∞ ( ∂ Ω) ≤ K as claimed. Up to extraction, the sequence ( v n ) converges uniformly to some lipschitz-continuous function v in Ω . In addition, v = 0 on ∂ Ω , and for any ϕ ∈ C ∞ c ( B ′ × ( − , , ´ Ω f ( v n ) ϕ dx → ´ Ω f ( v ) ϕ dx while ˆ Ω v n ( − (∆ x + ( µ ′ n ) ∂ y + µ ′′ n ∂ y )) ϕ dx = ˆ Ω v n ( − (∆ x + 1 n ∂ y )) ϕ dx → ˆ Ω v ( − ∆ x ϕ ) dx. In particular, the function w ( x ) = v ( x, ∈ H ( B ′ ) is a weak stable solution to − ∆ w = f ( w ) in B ′ .But so is u = − | x | ) . By uniqueness of the extremal solution, we must have w = u , which is impossible,since w is bounded. Hence, up to extraction, k u n k L ∞ ((Ω n ) ρn ) → + ∞ . Appendix A. Proposition 21.
Assume that α ∈ (0 , and N > . Let Ω be a bounded domain of R N and let f ∈ C ([0 , + ∞ )) be a nondecreasing and convex function. Assume that u ∈ H (Ω) is a stable solutionof (1) such that u ≥ a.e. in Ω .1) There exists a sequence ( ε n ) of real numbers in [0 , such that ε n ց and a sequence ( u n ) of functionsin H (Ω) ∩ C (Ω) such that u n is a stable weak solution to (68) − ∆ u n = (1 − ε n ) f ( u n ) in Ω ,u n − u ∈ H (Ω) , u n u a.e. on Ω , and (69) u n −→ u in H (Ω) , u n −→ u a.e. on Ω . Also, if we assume in addition that Ω is of class C , T is a C ,α open portion of ∂ Ω and u | T = 0 (inthe sense of the traces), then u n ∈ C ,α (Ω ′ ) for any domain Ω ′ ⊂⊂ Ω ∪ T (sufficiently small) and u n = 0 in T ′ .
14. Note that f ′ ( u ) is a nonnegative Lebesgue measurable function (by our assumptions on f ) and so the stabilityinequality (2) has a meaning.15. That is a function u n satisfying ´ Ω ∇ u n ∇ ϕ = ´ Ω (1 − ǫ n ) f ( u n ) ϕ , for all ϕ ∈ H (Ω) . EGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS. 17
2) Assume in addition that u ∈ H (Ω) . Then, the sequence ( u n ) can be chosen in H (Ω) ∩ C (Ω) . If weassume in addition that Ω is of class C , T is a C ,α open portion of ∂ Ω and u | T = 0 (in the sense ofthe traces), then u n ∈ C (Ω ′ ) for any domain Ω ′ ⊂⊂ Ω ∪ T (sufficiently small). In particular, if Ω is ofclass C ,α , then u n ∈ C (Ω) . Proof. i) We argue as in [3] and in subsection 3.2.2 of [10]. Nevertheless, our approach requires several nonstandard modifications due to the fact that we work in D ′ and that u is merely in H ( B ) (i.e., u doesnot have "zero boundary value").Given ǫ ∈ (0 , , define Φ ǫ : [0 , + ∞ ) → [0 , + ∞ ) by ˆ Φ ǫ ( t )0 dsf ( s ) = (1 − ǫ ) ˆ t dsf ( s ) Since Φ ǫ solves the initial value problem(70) ( Φ ′ ǫ ( t ) f ( t ) = (1 − ǫ ) f (Φ ǫ ( t )) , t > ǫ (0) = 0 , we see that Φ ǫ ∈ C ([0 , + ∞ )) is increasing, concave and satisfies < Φ ′ ǫ ( t ) < , Φ ǫ ( t ) t for all t > , Φ ′ ǫ (0) = 1 − ǫ ∈ (0 , , Φ ′′ ǫ (0) = − ǫ (1 − ǫ ) f ′ (0) f (0) Also, using the concavity on [0 , + ∞ ) of thefunction h ( t ) := ´ t dsf ( s ) we get that(71) f (Φ ǫ ( t )) C ( f ) ǫ (1 + t ) ∀ t > , where C = C ( f ) > is a constant depending only on f .Since u ∈ H (Ω) and (71) is in force, we have U ǫ = Φ ǫ ( u ) ∈ H (Ω) and so U ǫ is a weak supersolution to(72) ( − ∆ u ǫ, = (1 − ǫ ) f ( u ǫ, ) in Ω , u ǫ, − Φ ǫ ( u ) ∈ H (Ω) , while v = 0 is a weak subsolution to (72). In addition, we have < U ǫ a.e. on Ω , since f > . Therefore bythe (standard) method of sub and supersolution in H we obtain a stable weak solution u ǫ, ∈ H (Ω) of (72) such that < u ǫ, U ǫ a.e. on Ω . Furthermore, from (71) we get(73) f ( u ǫ, ) f ( U ǫ ) C ( f ) ǫ (1 + u ) and so f ( u ǫ, ) ∈ L (Ω) . The latter implies u ǫ, ∈ H loc (Ω) by elliptic regularity, hence(74) u ǫ, ∈ L ploc (Ω) ∀ p < NN − p ∞ if N , p < ∞ if N = 4) by Sobolev imbedding.In what follows, for any integer j > , we shell denote by Φ jǫ the composition of Φ ǫ with itself j times( Φ ǫ = Id .)
16. That is it satisfies ´ Ω ∇ U ǫ ∇ ϕ > ´ Ω (1 − ǫ ) f ( U ǫ ) ϕ , for all ϕ ∈ H (Ω) , ϕ > . Indeed, in view of the above propertiesof Φ ǫ we can extend it to a C function on the entire real line R (still denoted by Φ ǫ ) such that Φ ǫ is nondecreasing andconcave, Φ ′ ǫ is nonnegative and bounded on R . Then we can apply a variant of Kato’s inequality (see e.g. Lemma 3.2.1 in[10]) to get that U ǫ is a supersolution in D ′ (Ω) . A standard density argument and Fatou’s Lemma then give the desiredconclusion.17. This solution is obtained by using the standard method of monotone iterations in H applied to the sequence ( v k ) k > defined by − ∆ v k +1 = (1 − ǫ ) f ( v k ) in Ω , v k +1 ∈ { v ∈ H (Ω)) : v − Φ ǫ ( u ) ∈ H } := H ǫ ( u ) and starting with v =Φ ǫ ( u ) ∈ H ǫ ( u ) , the supersolution. Note that the sequence is well-defined in H ǫ ( u ) and satisfies v k +1 v k Φ ǫ ( u ) a.e. on Ω thanks to f ′ > and since f (Φ ǫ ( u )) ∈ L ( B ) by (71). Furthermore, the stabilty of u ǫ, comes from the stabilty of u and the fact that f ′ is positive and nondecreasing. Indeed, ∀ ϕ ∈ C c (Ω) we have ´ Ω |∇ ϕ | > ´ Ω f ′ ( u ) ϕ > ´ Ω ( f ′ ( U ǫ ) ϕ > ´ Ω ( f ′ ( u ǫ, ) ϕ > ´ Ω ((1 − ǫ ) f ′ ( u ǫ, ) ϕ . Now we can repeat the same construction to find a stable weak solution u ǫ, ∈ H (Ω) to(75) ( − ∆ u ǫ, = (1 − ǫ ) f ( u ǫ, ) in Ω , u ǫ, − Φ ǫ ( u ) ∈ H (Ω) , such that < u ǫ, Φ ǫ ( u ǫ, ) u ǫ, u a.e. on Ω . Here we have used Φ ǫ ( u ǫ, ) ∈ H (Ω) as supersolutionand again v = 0 as subsolution. Also note that Φ ǫ ( u ) Φ ǫ ( u ǫ, ) on ∂ Ω in the sense of H (Ω) , since [Φ ǫ ( u ) − Φ ǫ ( u ǫ, )] + ∈ H (Ω) . In particular, by (71), f ( u ǫ, ) C ( f ) ǫ (1 + u ǫ, ) ∈ L ploc (Ω) for any p inthe range (74), and thus u ǫ, ∈ L qloc (Ω) for all q < NN − ( q ∞ if N , q < ∞ if N = 8 ). Also notethat < u ǫ, Φ ǫ ( u ǫ, ) Φ ǫ (Φ ǫ ( u )) = Φ ǫ ( u ) Φ ǫ ( u ) a.e. on Ω .By iteration, we find that if k = h N i + 1 , the solution u ǫ,k ∈ H (Ω) to(76) ( − ∆ u ǫ,k = (1 − ǫ ) k f ( u ǫ,k ) in Ω , u ǫ,k − Φ kǫ ( u ) ∈ H (Ω) , is locally bounded (hence of class C inside Ω ) and also satisfies < u ǫ,k Φ ǫ ( u ǫ,k − ) Φ kǫ ( u ) u a.e.on Ω .Since ǫ ∈ (0 , is arbitrary we have proved that, for every δ ∈ (0 , (choose δ = 1 − (1 − ǫ ) k ) thereexists a nonnegative stable weak solution u δ ∈ H (Ω) ∩ C (Ω) to(77) ( − ∆ u δ = (1 − δ ) f ( u δ ) in Ω , u δ − Φ kδ ( u ) ∈ H (Ω) . Since u δ Φ kδ ( u ) u a.e. on Ω by construction, we get k u δ k L (Ω) k Φ kδ ( u ) k L (Ω) k u k L (Ω) andalso that Φ kδ ( u ) −→ u in L (Ω) by the dominated convergence theorem (recall that Φ δ ( t ) −→ t for all t > and that Φ ǫ is a contraction on R + ). Moreover, by choosing Φ kδ ( u ) − u δ ∈ H (Ω) as test functionin the weak formulation of (77) we obtain ˆ Ω ∇ u δ ∇ (Φ kδ ( u ) − u δ ) = ˆ Ω (1 − δ ) f ( u δ )(Φ kδ ( u ) − u δ ) > since Φ kδ ( u ) − u δ > a.e. on Ω and f > . Therefore we deduce that ´ Ω |∇ u δ | ´ Ω ∇ u δ ∇ Φ kδ ( u ) which leads to k∇ u δ k L ( B ) k∇ Φ kδ ( u ) k L ( B ) by Young’s inequality. On the other hand ∇ Φ kδ ( u ) = (cid:16) Q k − j =0 Φ ′ δ (Φ ( j ) δ ( u )) (cid:17) ∇ u , which entails k∇ u δ k L (Ω) k∇ Φ kδ ( u ) k L (Ω) k∇ u k L (Ω) . Therefore(78) k u δ k H (Ω) k Φ kδ ( u ) k H (Ω) k u k H (Ω) . In particular the families ( u δ ) and (Φ kδ ( u )) are bounded in H (Ω) and therefore, we may and do supposethat (up to subsequences) u δ ⇀ v in H (Ω) , u δ −→ v in L (Ω) , u δ −→ v a.e. on Ω and Φ kδ ( u ) ⇀ V in H (Ω) , Φ kδ ( u ) −→ V in L (Ω) , Φ kδ ( u ) −→ V a.e. on Ω , for some v, V ∈ H (Ω) , as δ → . Fromthose properties we get V = u (recall that Φ kδ ( u ) −→ u in L (Ω) ) and also that v is a solution of − ∆ v = f ( v ) in Ω . Also v is stable thanks to u δ −→ v a.e. on Ω , the positivity and the continuity of f ′ and Fatou’s Lemma. On the other hand, the weak convergence of ( u δ ) and (Φ kδ ( u )) in H (Ω) and (77)imply u δ − Φ kδ ( u ) ⇀ v − u in H (Ω) . Finally we get that u δ −→ u in H (Ω) , since u δ ⇀ u in H (Ω) and lim sup k u δ k H (Ω) k u k H (Ω) by (78). Since both u and v are stable solutions in H (Ω) and v u a.e. on Ω , we deduce from Theorem 3 that either u = v in Ω or u > and u ∈ C ∞ (Ω) . In the first casethe desired conclusion follows by taking ǫ n = δ n , where ( δ n ) is any sequence in (0 , such that δ n ց and u n = u δ n , while in the second one it is enough to take ǫ n = 0 , u n = u for every n > .The last claim of item i) then follows by standard elliptic theory.ii) If f (0) > , the conclusion follows from item i). If f (0) = 0 , either u ≡ or u is a positive firsteigenfunction of − ∆ with homogeneous Dirichlet boundary conditions, by Theorem 18. In both cases thesmoothness of u up to (a portion of the) boundary follows from the elliptic regularity, since ∂ Ω is smoothenough. To conclude it is enough to take ǫ n = 0 , u n = u for every n > . (cid:3) EGULARITY AND SYMMETRY FOR SEMILINEAR ELLIPTIC EQUATIONS IN BOUNDED DOMAINS. 19
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