Superlinear critical resonant problems with small forcing term

Abstract

We prove the existence of solutions of a class of quasilinear elliptic problems with Dirichlet boundary conditions of the following form

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} Lu= g(u) -f &{} \hbox { in } \quad \Omega ,\\ u\in X, \end{array} \right. \end{aligned}$$

where $\Omega \subset \mathbb R^N$ is a bounded domain, $N\ge 2$, the differential operator is $Lu= -\hbox {div}( |\nabla u|^{p-2}\nabla u )-\lambda _1 |u|^{p-2}u$ with $X=W^{1,p}_0(\Omega )$ or $Lu= \Delta ^2u -\lambda _1 u$ with $X=H^2_0(\Omega )$, the nonlinearity is given by $g(u)=(u^+ )^q$ or $g(u)=|u|^{q-1}u$ i.e. it is a superlinear, at most critical, term and $f$ is a small reaction term. We give an abstract formulation for which solutions are found by minimization on an appropriate subset of the Nehari manifold associated to our problem. Our method can be also applied to other related problems involving indefinite weights.