Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents

Abstract

The main results of this paper establish, via the variational method, the multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents under the presence of symmetry. The concentration-compactness principle allows to prove that the Palais-Smale condition is satisfied below a certain level.

Multiplicity of solutions for quasilinear elliptic systems with critical growth

“…Note that (H 2 ) is a version of the Ambrosetti-Rabinowitz condition which was considered in [29] for the scalar problem. This hypothesis and the Concentration-Compactness Principle of Lions [22] allow us to verify that the associated functional satisfies the Palais-Smale condition below a fixed level for appropriated values of µ 1 , µ 2 > 0.…”

“…This hypothesis and the Concentration-Compactness Principle of Lions [22] allow us to verify that the associated functional satisfies the Palais-Smale condition below a fixed level for appropriated values of µ 1 , µ 2 > 0. Observe that we may not guarantee the existence of nontrivial solutions for (1.1) when µ 1 = µ 2 = 0 since, in this case, the associated functional may not satisfy any compactness condition (see example in [29]). The hypothesis (H 3 ) and the existence of Schauder bases for W 1,p 0 (Ω) and W 1,q 0 (Ω) provide the local geometric condition required by the Symmetric Mountain Pass Theorem.…”

Positive Solutions of Quasilinear Elliptic Equations with Critical Orlicz-Sobolev Nonlinearity on RN

“…In the case of bounded W, positive solutions of (1.2) are obtained by Brézis and Nirenberg [10], Guedda and Véron [17], García Azorero and Peral Alonso [15], BenNaoum, Troestler and Willem [7], and Silva and Xavier [21]. Further, Benci and Cerami [6], Gonçalves and Alves [16], Ambrosetti, Garcia Azorero and Peral [2,3], and Silva and Soares [20] have studied the case of W ¼ R N .…”

MULTIPLICIDADE DE SOLUÇÕES PARA UMA CLASSE DE PROBLEMAS QUASELINEARES CRÍTICOS EM R<SUP>N</SUP> ENVOLVENDO FUNÇÃO PESO COM MUDANÇA DE SINAL

“…For more general results in bounded domains see e.g. the papers by Ambrosetti et al (1996); Birindelli e Demengel (2004); Pacella et al (1997) de Figueiredo et al (2006); Silva e Xavier (2003); Azore ro et al (2000) and their references.…”