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, (Formula presented.) xwhere Ω ⊂ Rn is a bounded domain with C2-boundary and 1 OpenSPiltSPi qOpenSPiltSPi 2 OpenSPiltSPi p. As a consequence of our results we shall show that, for each pCloseSPigtSPi 2 , there exists μ∗CloseSPigtSPi 0 such that for each μ∈ (0 , μ∗) 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