Multiplicity Results of Critical Local Equation Related To the Genus Theory

Abstract

Using variational methods, Krasnoselskii's genus theory and symmetric mountain pass theorem, we introduce the existence and multiplicity of solutions of a parameteric local equation. At first, we consider the following equation( -div[a(x, | backward difference u|) backward difference u] = mu(b(x)|u|s(x)-2 - |u|r(x)-2)u in S2, u = 0 on partial differential S2, where S2 subset of RN is a bounded domain, mu is a positive real parameter, p, r and s are continuous real functions on S2 over bar and a(x, xi) is of type |xi|p(x)-2. Next, we study boundedness and simplicity of eigenfunction for the case a(x, | backward difference u|) backward difference u = g(x)| backward difference u|p(x)-2 backward difference u, where g is an element of L infinity(S2) and g(x) >= 0 and the case a(x, | backward difference u|) backward difference u = (1 + backward difference u|2) p(x)-2 2 backward difference u such that p(x) equivalent to p.