A Perturbation Method in Critical Point Theory and Applications

Abstract

This paper is concerned with existence and multiplicity results for nonlinear elliptic equations of the type $- \Delta u = {\left | u \right |^{p - 1}}u + h(x)$ in $\Omega , u = 0$ on $\partial \Omega$. Here, $\Omega \subset {{\mathbf {R}}^N}$ is smooth and bounded, and $h \in {L^2}(\Omega )$ is given. We show that there exists ${p_N} > 1$ such that for any $p \in (1, {p_N})$ and any $h \in {L^2}(\Omega )$, the preceding equation possesses infinitely many distinct solutions. The method rests on a characterization of the existence of critical values by means of noncontractibility properties of certain level sets. A perturbation argument enables one to use the properties of some associated even functional. Several other applications of this method are also presented.