Eigenvalue Problems For Singular Quasilinear Elliptic Equations With Weights

In this paper, the existence of nontrivial solutions of three classes of quasilinear elliptic equations resonance problems with high eigenvalues in a weighted Sobolev space are proved by using Galerkin method, Brouwer’s theorem and a new weighted compact Sobolev-type embedding theorem established by V. L. Shapiro. The paper is organized as follow.In chapter one, we mainly introduce the developing background of the nonlinear partial differential equations, the significance of the issue and present stage of research. State the problems what we have solved in this paper. Furthermore, we introduce the main method, basic theorems and some fundamental lemmas in order to prove the theorem. We also give some assumptions about the spaces, functions and operators.In chapter two, we consider the existence of nontrivial solutions on the weighted singular quasilinear elliptic equation Where We generalize the similar research about the solution of the first eigenvalue to high eigenvalue. On the other hand, we get the space structure more complicated than before.In chapter three, on the basis of chapter two, we prove the existence of nontrivial solutions of the weighted singular quasilinear elliptic equation resonance problem In this chapter the operator Ｌ is same as the chapter two. The complexity of the problem is generated by the nonlinear terms and high eigenvalue.In chapter four, we establish the relationship which is called near-related between quasilinear operator Μ. and the linear operator Ｌ. And we consider the existence of solutions of a kind of weighted singular quasilinear elliptic equations under the condition that they meet the relationship which is established above where...