On Some Elliptic Eigenvalue Problems

In this paper, we discuss the continuous solutions of an elliptic equation by minimax method, mountain pass theorem and linking theorem, which are the basic method of variational principle. The structure of this paper goes as follow:In the introduction, we review the application background for these problems and learn the practical application background for the problem we will solve.In chapter one, there are some necessary basic knowledge and important basic lemma for solving the elliptic problem.In chapter two, there is discussion about the function and limitation of (AR) condition in solving elliptic problems. Taking the following equation for example:-Δu+a(x)u=f(x,u), x∈Ω, u|(?)Ω=0. Consider the different cases for searching the weak solution of the preceding equation with (AR) condition and without (AR) condition. For the cases, we will see the mountain pass geometry structure under the(PS)c condition and (C)c condition.In chapter three, since the study of many elliptic problem bases on the elliptic eigenvalue equation-Δu=λg(x,u), x∈Ω, u|(?)Ω=0(Ⅰ) which we can find from the chapter two, when we want to study the properties of solutions for elliptic equation, we consider to study the continuity of solutions respect to the parameter r for the elliptic eigenvalue equation with constraintThen we obtain the main result for this chapter:elliptic eigenvalue problem (Ⅰ) exist two branches of solutions.In chapter four, we conclude and analyse the function and limitation of (AR) condition in the application of mountain pass lemma in solving elliptic equation, and have a assumption on finding the branch of sign-changing solution for elliptic equation.