Existence Of Ground State And Positive Solutions For Nonlinear Elliptic Equation

Variational method of the differential equation is transformed into a question of finding the critical points of corresponding function in an appropriate Banach space. When studying nonlinear elliptic equations, the corresponding function may neither have upper nor lower bound. In this article, the existence of ground state solution of superlinear p-Laplace elliptic equation and positive solution of asymptotically linear elliptic equation without (AR) condition is studied with the help of using a variant version of the Mountain Pass Theorem, the Variational Principle, and Strong Maximum Principle.(1) Consider a superlinear p-Laplace elliptic equation conditions of superlinear, a bounded (PS) sequence of the corresponding functional I is obtained by using Ekeland variational principle, which satisfies the condition of Mountain Pass Theorem, and then get the bounded Cerami sequence of I. At last, the bounded Cerami sequence has a subsequence, which converges strongly to a ground state solution of equation that is proved.(2) Consider the asymptotically linear elliptic equationWhere A is Laplace operator. Assume that the elliptic equation do not satisfy the condition of (AR), applying a variant version of the Mountain Pass Theorem to get a bounded (PS) sequence of the corresponding functional 1, and it converges to a positive solution of the elliptic equation. Then get the bounded Cerami sequence of 1, which converges strongly to subsequence by using Ekeland variational principle. At last, the subsequence converges strongly to the other positive solution of equation is proved by using Strong Maximum Principle.