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

We shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form,

$$\begin{aligned} \left\{ \begin{array}{ll} -\,\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, &{}\quad x \in \Omega \\ u=0, &{}\quad x \in \partial \Omega \end{array} \right. \end{aligned}$$

(1)

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.