Mountain Pass Lemma And Existence Of The Solutions For Partial Differential Equations With Asymptotically Linear Property

This paper is concerned with the existence of the solutions forthree types of nonlinear partial di?erential equations with asymptotically linearproperty at infinity.At first we study the existence of the solutions of Dirichlet problems forthe asymptotically linear elliptic equations with nonnegative potentialBy defining a constraint variational problem and using the improved mountainpass theorem, the existence of positive solutions of the Dirichlet problems for(0-1) is obtained without (AR) condition.Next we investigate the existence of the solutions for the asymptoticallylinear elliptic equations with unbounded potentialBy using the interpolation estimate, the existence of the solutions for a new con-strained variational problem is obtained. Furthermore, by using the deformedmountain pass theorem, the existence of nontrivial solutions for (0-2)is gotten.At last, we study the existence of the travelling wave solutions for K-PEquationwhere f(x,y,w) is asymptotically linear with respect to w at infinity. Bydefining a constraint variational problem and using an improved mountain pass theorem, the existence of the travelling wave solutions for (0-3) is obtainedwithout (AR) condition.