The Multiple Positive Solutions For A Class Of Superlinear Elliptic Equations

In this paper, we investigate the multiple positive solutions for aclass of superlinear elliptic equations. Studying the nonlinear elliptic equationsby Minimax methods, many scholars in the field always assume that thenonlinearity f(x,u) satisfies the famous Ambrosetti-Rabinowitz condition, whichis called (AR) condition for short. (AR) condition guarantees that all of the(PS) sequences are bounded of the functional for the studied elliptic equation.That is the important premise of Minimax methods. However, there are manyfunctions in application which do not satisfy the (AR) condition. Then, ourmain work in this paper is to study the multiple positive solutions for a classof superlinear elliptic equations without the assumption that the nonlinearity fdoesn't satisfy the Ambrosetti-Rabinowitz condition. The method we use is amodification of the Fountain Theorem. We obtain the result through the proofof variational functional satisfying the Cerami's condition, and we generalizesome known results.