Multiplicity results for elliptic problems with super-critical concave and convex nonlinearties

Abstract

We shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form, 1 $$\begin{aligned} \left\{ \begin{array}{ll} -\,\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, &{}\quad x \in \Omega u=0, &{}\quad x \in \partial \Omega \end{array} \right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^n$$ is a bounded domain with $$C^2$$ -boundary and $$1<q< 2<p.$$ As a consequence of our results we shall show that, for each $$p>2$$ , there exists $$\mu ^*>0$$ such that for each $$\mu \in (0, \mu ^*)$$ problem (1) has a sequence of solutions with a negative energy. This result is already known for the subcritical values of p. In this paper, we shall extend it to the supercritical values of p as well. Our methodology is based on a new variational principle established by one of the authors that allows one to deal with problems beyond the usual locally compactness structure.