The Existence Of Solutions For An Nonlinear Elliptic Equation

In this paper, we discuss the existence of solutions for some nonlinear elliptic equa-tions by using Mountain pass lemma, critical point theory and linking theory.This paper is organized as follows. In the first chapter, the developments of nonlinearelliptic equations, the theorems and important inequalities which will be used in the paper,and the main idea of the paper will be presented.In the second chapter, by establishing a new Hilbert space, we discuss the existenceof nontrivial solutions of a class of nonlinear elliptic equations with Hardy potential inthe new Hilbert space. We prove the existence of nontrivial solution in the new space byusing the Mountain pass theorem with the PS condition.In the third chapter, by constructing a new space, we discuss the existence of sign-changing solution of a fourth-order nonlinear elliptic equation in the new Hilbert space.The existence of sign-changing solution for fourth-order nonlinear elliptic equation isobtained by using linking theorem.In the fourth chapter, by constructing of a new function space, we discuss the ex-istence of sign-changing solution of a fourth-order nonlinear elliptic equation in the newHilbert space. we prove that this equation which has infinitely many sign changing solu-tions when f(x, t) satisfies certain conditions.