The Existence Of Solutions For Nonlinear Elliptic Equations

In this paper, making use of some conventional theory tools such as Compact imbeddingtheorem, Mountain pass lemma and Nehari skill, we focus mainly on the existence of solutionsfor some nonlinear elliptic equations. This paper consists of four chapters:In chapterⅠ, the historical background, research developments, main methods and achieve-ments on nonlinear elliptic equations are summarized. Then the main conclusions of this paperare simply introduced.In chapterⅡ, we study the existence of nontrivial solution for superlinear elliptic equa-tions with Hardy potential |x1|2 in RN(N≥3). We discuss the problem in a new Hilbert spaceH which is the completion of C0∞(?) with respect to the new norm. Furthermore using Ceramicondition and the Mountain pass lemma, we prove that there is a nontrivial solution for theequations in the new space H.In chapterⅢ, let 0∈? ? RN(N≥3)be a bounded domain with smooth boundary.We discuss the existence of positive solutions near zero for a class of weighted nonlinear el-liptic equations ? div with Dirichlet boundary condi-tion. Then we obtain multiple positive solutions for above equations involving critical weightedSobolev-Hardy exponents.In chapterⅥ, we investigate a class of elliptic equations involving weighted Sobolev-Hardy terms. A detailed analysis on the PS sequence of the variational functionals correspond-ing to the equations is given and a local compactness result is obtained. Applying this com-pactness result, the Mountain pass lemma and the Strong Maximum Principle, we prove theexistence of the above equations under certain conditions.