Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term

Abstract

We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain Ω ⊂ R N \Omega \subset {{\mathbf {R}}^N} \[ − div ⁡ ( | ∇ u | p − 2 ∇ u ) = | u | p ∗ − 2 u + λ | u | q − 2 u , λ > 0 , - \operatorname {div} (|\nabla u{|^{p - 2}}\nabla u) = |u{|^{{p^{\ast }} - 2}}u + \lambda |u{|^{q - 2}}u,\qquad \lambda > 0, \] where p ∗ {p^{\ast }} is the critical Sobolev exponent, and u | δ Ω ≡ 0 u{|_{\delta \Omega }} \equiv 0 . By using critical point methods we obtain the existence of solutions in the following cases: If p > q > p ∗ p > q > {p^{\ast }} , there exists λ 0 > 0 {\lambda _0} > 0 such that for all λ > λ 0 \lambda > {\lambda _0} there exists a nontrivial solution. If max ( p , p ∗ − p / ( p − 1 ) ) > q > p ∗ \max (p,{p^{\ast }} - p/(p - 1)) > q > {p^{\ast }} , there exists nontrivial solution for all λ > 0 \lambda > 0 . If 1 > q > p 1 > q > p there exists λ 1 {\lambda _1} such that, for 0 > λ > λ 1 0 > \lambda > {\lambda _1} , there exist infinitely many solutions. Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.