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 − Δ u = | u | p − 1 u + h ( x ) - \Delta u = {\left | u \right |^{p - 1}}u + h(x) in Ω , u = 0 \Omega ,\,u = 0 on ∂ Ω \partial \Omega . Here, Ω ⊂ R N \Omega \subset {{\mathbf {R}}^N} is smooth and bounded, and h ∈ L 2 ( Ω ) h \in {L^2}(\Omega ) is given. We show that there exists p N > 1 {p_N} > 1 such that for any p ∈ ( 1 , p N ) p \in (1,\,{p_N}) and any h ∈ L 2 ( Ω ) 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.