The Existence Problems Of Some Semilinear Elliptic Equations

In this paper, we study the existence of solutions to two classes of elliptic partial differential equations under different conditions.In the first chapter, we review the background of the problem, and introduce the variational method and the Kirchhoff equation:In the second chapter, we introduce some basic knowledge and basic lemma of Sobolev space;In the third chapter, the Kirchhoff type equation with the radial potential is considered: where a> 0, b> 0 and V, Q satisfied the condition (V), (Q). By using the mountain pass theorem, the above equation at least possesses one nontrivial solution under different conditions.In the fourth chapter, the Kirchhoff type equation is considered: where ? is a smooth bounded domain in RN(N?1),b>0, and f(x,u):?×R is a continuous real function and satisfies the subcritical growth condition. By using the mountain pass theorem, the above equation at least has one positive solution under different conditions.