Existence Of Solutions For Quasi-linear Elliptic Equations

In this paper, we mainly study the existence of nontrivial solutions for two classesof quasi-linear elliptic equations. One is the perturbed Hardy-Sobolev operator withsingular weight, and the other is N-Laplacian equations with critical growth. A newlinking theorem of the critical point theory is used as a major tool.This paper consists of four chapters. In chapter1, we lay out the source of theproblems in detail, and state the main results of this paper in the end.In chapter2, we sketch some preliminaries, collect two kinds of index theoriesused in p-Laplace equations, and summarize the history of the linking theorem. At last,we give a new linking theorem which is used in our paper.In chapter3, we make a perturbed function for Hardy term of the associatedp-Laplace equations. As the weight function with some specific singularity and thegrowth term satisfy subcritical growth, the problem admits a nontrivial solution. Firstly,we discuss the properties (e.g. existence, isolation of the first eigenvalue problem,several presentations of the second eigenvalue and divergent eigenvalue sequence) ofthe eigenvalue problem under the double action of perturbed function and weightfunction. Secondly, applying the above results, we construct the cone structurescoincided with the linking theorem to obtain the nontrivial solution of the problem forany positive eigenvalue parameter.In chapter4, the nonlinear term is defined by the Trudinger-Moser inequality. Forthis problem, the cone structure of the above is not suitable. We need an ingeniousinvestigation for the corresponding eigenvalue problem. By the concentration techniqueon points moving to the boundary, regularity of the solutions of the associatedeigenvalue problem, and constructing a modified Moser function, we obtain theexistence of linking structure for the problem. At last, the functional associated theproblem satisfies the compact condition under some energy level, and we obtain theexistence of the nontrivial solutions of the problem for eigenvalue parameter large thanthe first eigenvalue and not equal to the eigenvalues defined by the2-cohomologicalindex by linking theorem.