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 <!-- MATH $- \Delta u = {\left| u \right|^{p - 1}}u + h(x)$ --> in <!-- MATH $\Omega ,\,u = 0$ --> on <!-- MATH $\partial \Omega$ --> . Here, <!-- MATH $\Omega \subset {{\mathbf{R}}^N}$ --> is smooth and bounded, and <!-- MATH $h \in {L^2}(\Omega )$ --> is given. We show that there exists 1$"> such that for any <!-- MATH $p \in (1,\,{p_N})$ --> and any <!-- MATH $h \in {L^2}(\Omega )$ --> , the preceding equation possesses infinitely many distinct solutions.