On The Solutions Of Asymptotically Linear And Superlinear Elliptic Equations

The research of the differential equation can go back to the previous century. Variational method of the differential equation is one of the main methods to research the elliptic equations. The basic idea is to solve nonlinear elliptic equations solution for the problem that boils down to the point of a functional. And one method of finding the critical point is Mountain Pass Lemma, but in order to apply the lemma, (AR) condition is required. However, in many cases, (AR) condition dose not satisfy. Then, our main work is applying a variant version of the Mountain Pass Theorem to study the existence of the solutions of asymptotically linear elliptic equations and superlinear elliptic equations without (AR) condition.(1) This paper discusses the asymptotically Kirchhoff-type elliptic equation where Q is a bounded domain in RN with smooth boundary (?)?. First, we prove that the energy functional I of the equation satisfies the Mountain Pass geometric properties and get the (P-S) sequence of I; Second, we prove that the (P-S) sequence of I is bounded; At last, we prove that the bounded (P-S) sequence has a subsequence, which converges strongly to a positive solution of equation,(2) This paper consider the Dirichlet problem for the elliptic equation Where ? is a bounded domain in RN with smooth boundary (?)?.-?p is P-Laplace operator, and -?p(u)=-div(|Du|p-2Du). First, we prove that the energy functional I of the equation satisfies the Mountain Pass geometric properties and get the Cerami sequence of I; Second, we prove that the Cerami sequence of I is bounded; At last, we prove that the bounded Cerami sequence has a subsequence, which converges strongly to a nontrivia weak solution of equation.