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{equation}\label{con-c} \left \{ \begin{array}{ll} -\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, & x \in \Omega\\ u=0, & x \in \partial \Omega \end{array} \right. \end{equation} 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 $\mu \in (0, \mu^*)$ this problem has a sequence of solutions with a negative energy. This result was 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.