Multiple Solutions For The Quasi-linear Elliptic Equations

In recent years, there has been a growing interest towards variational differential problems, involving p-Laplacian, which arise naturally in various physical contexts, for instance, in the study of non-Newtonian fluids and in the study of elasticity problems. It is well known that Morse theory in Hilbert spaces has been largely used to obtain multiplicity results of solutions for semilinear elliptic equations, having a variational structure. A first problem is that the classical definition of nondegenerate critical point introduced in a Hilbert space, i.e.,f"(u) is an isomorphism from Xto the dual space x*, does not seem very reasonable, since there are several examples of Banach spaces which are not isomorphic to their dual spaces.In this paper, we consider the quasi-linear elliptic boundary value problem whereΔpw=div(|Δu|p-2Δu),l<p<∞, and Ωis a bounded domain in R"(n>1)with smooth boundary δΩ.The first part of this paper is to study some qualitative results of the critical groups of an isolated critical point for this p-Laplacian equation and the existence of sign changing solutions of the equation are given by Morse theory.In the sceond part of this paper, by Morse theory we will compute the critical groups at zero for this equation, and we assume that f is resonant at zero for the spectrum of Δ-in H01(Ω). As an application of this critical groups estimates, some multiplicity results are also given.