An existence result for nonlinear elliptic problems involving critical Sobolev exponent

نویسندگان

  • A. CAPOZZI
  • D. FORTUNATO
  • G. PALMIERI
چکیده

In this paper we consider the following problem: where Q c Rn is a bounded domain and We prove the existence of a nontrivial solution of (1) for any ~, > 0, RESUME. Soient Q un sous-ensemble ouvert borne de Rn et À un nombre positif, le but de cette note c’est de montrer que le probleme suivant : admet, au moins, une solution non triviale, si r~ > 4. Work supported by G. N. A. F. A. or C. N. R. and by Ministero Pubblica Istruzione (fondi 40 % e 60 %). Annales de l’Institut Henri Poincaré Analyse non linéaire Vol. 2, 0294-1449 85/06/463 -8 2,80; C Gauthier-Villars 464 A. CAPOZZI, D. FORTUNATO AND G. PALMIERI 0 . INTRODUCTION Let Q c (~n, n > 3, be an open bounded set with smooth boundary. Consider the problem where ~, is a real parameter and 2* = 2n/(n 2) is the critical Sobolev exponent for the Sobolev embedding Ho(~) c~ LP(Q). The solutions of (0.1) are the critical points of the energy functional Since the embedding Ho(S~) ~ is not compact the functional does not satisfy the Palais-Smale condition in the energy range ] oo, + oc [ (cfr. remark 2 . 3 of [4 ]). Moreover if £ 0 and Q is starshaped (0.1) has only the trivial solution (cf. [6 ]). Recently Brezis and Nirenberg in [2] ] have proved that if n > 4 and 0 ~. ~,1 (~~ 1 is the first eigenvalue of A) then (0.1) has a positive solution. In [4 ] Cerami, Fortunato and Struwe have obtained multiplicity results for (0.1) in the case in which £ belongs to a suitable left neighbourhood of an arbitrary eigenvalue of 0394 (cf. also [3 ]). In this paper we prove the following theorem: THEOREM 0.1. If n > 4 the problem (0.1) possesses at least one tion trivial solution for any ~, > 0. A weaker result related to theorem 0 .1 has been announced in [5]. We observe that if n = 3 and Q is a ball, Brezis and Nirenberg [2 ] have proved that the problem (0.1) does not have nontrivial radial solutions if 0 03BB 03BB1 4. 1. SOME PRELIMINARIES Let [ [ ~p denote respectively the norms in and LP(Q) (1 -p x), and let denote the best constant for the embedding L2*{S~). The following lemma shows that f;~ satisfies a local P. S. condition. Annales de l’Institut Henri Poincaré Analyse non linéaire 465 CRITICAL SOBOLEV EXPONENT LEMMA I .1. For any i~ E ~ the functional f;~ (see (0. 2}) satisfies the Palais-Smale condition in x, in the follow~ing sense: If c ~ and is a sequence in such that n J as m -~ ~o -~ c, 0 strongly in H ’ (S2), then s~ (P. S.) { contains a subsequence converging strongly in The proof of this lemma is in [2 ] and in [4 ]. We recall that a deeper compactness result has been proved in [7 ]. We recall a critical point Theorem (cf. [l, Theorem 2 . 4 ]) which is a variant of some results contained in [0]. THEOREM 1. 2. Let H be a real Hilbert space and f E C1(H, f~) be a fimctional satisfying the following assumptions: f u), f ’(0) = 0 for any i~ E H ( , f ; ) there exists ~3 > 0 such that f satisfies (P. S. ) in ]0, j3 [ ( f3 ) there exist two closed subspaces V, H and positive constants p, ~ such that (i) f(u) ~3 for any u E W (ii) feu) > b for any u E V, ~) u ~~ p (iii) codim V + oo. Then there exist at least m pairs of critical points, with m = dim (V n W) codim (V + W) . 2. PROOF OF THEOREM 0.1 Uur aim is to define two suitable closed subspaces V and W, with V n W ~ ~ 0 ~ and V + W = H, such that f’; satisfies the assumptions and f3) of Theorem 1.2 with 1 n In the sequel we denote the eigenvalues and by M(i.~) the corresponding eigenspaces. Given i > 0, we set where the closure is taken in Ho(~). If r > 0 we set

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تاریخ انتشار 2017