Existence And Multiplicity Of Solutions For The Semilinear Elliptic Dirichlet Problem At Resonance Of The First Eigenvalue

In this paper we study the existence and multiplicity of solutions for the semilinear elliptic Dirichlet problem at the first eigenvalue.Our purpose in this paper is that 1) studying the efficiency and succinctly of the Saddle Point Reduction at finding solutions with saddle point character;2)finding single or multiplicity results of problem (1.1).The main results of this dissertation are as follows: 1) First, we give the remarks on the cited reference and relate the developments of the existence of solutions of problem (1.1); 2) In the next section , the fundamental knowledge are given; 3) In the third section, the existence of solution is obtained by using the reduction method and (LL) condition . 4) In the section four .we obtain the multiplicity of solution for the problem (1.1) by using the reduction method and the critical point theory.