Solutions Of Two Types Of Elliptic Equations With Concave And Convex Nonlinear Terms

In this paper, we mainly consider two problems. Firstly, we consider the following nonlinear Schrodinger-Poisson system:where ? and ? are parameters, V ? C(R3,R), f, ? C(R3×R,R). The nonlinearity?g(x,u) + ?f(x,u) is odd in u and may involve a combination of concave and convex terms. A special case of the nonlinearity is ?|u|q-2u+?|u|P-2u with 1<q<2, 4<p<6.Under some suitable assumptions, we proved the existence of infinitely many solutions with high energy and negative energy by using Fountain theorem and Dual Fountain theorem in chapter two.Secondly, We consider the following nonlinear Kirchhoff type problem of the form where ? (?) R3 is a bounded domain with smooth boundary (?)? and a > 0, b ? 0. The nonlinearity ?g(x,u) + f(x,u) may involve a combination of concave and convex terms.Under some suitable conditions on f,g?C((?)×R,R) and ? ? R,we prove the existence of infinitely many high energy solutions of (K) by using Fountain theorem in chapter three. In particular, by using the method of invariant sets of descending flow, we prove that (K) has at least one sign-changing solutions.