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} -Δu =|u|^{p-2} u+μ|u|^{q-2}u, & x \in Ω\\ u=0, & x \in \partial Ω\end{array} \right. \end{equation} where $Ω\subset \mathbb{R}^n$ is a bounded domain with $C^2$-boundary and $1 2$, there exists $μ^*>0$ such that for each $μ\in (0, μ^*)$ 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.