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