Multiplicity Of Positove Solutions For A Class Of Elliptic Equations In R`n

The elliptic problems are focused in universality for their deep physical circumstance. In recent ten years, the positive solution of the elliptic problems with critical Sobolev exponent is one of the heated discussions hi this field. In 1973, Amborosetti & Rabinowitz proposed Mountain Pass Lemm which is well known for all of us, they applied it to solve the existence of the positive solutions of regular semi-linear elliptic equations with Dirichlet boundary value, Professer Shen Yiaotian extended the work in this area, he first applied it to solve the existence of nontrivial solution of the quasi-linear equation; and got a great deal of results in concerning with the existence of nontrivial solutions and multiply solutions of the quasi-linear elliptic equation [12]-[14]. At present, to apply the Mountain Pass Lemma corresponded with variation method to study the elliptic partial differential equations is the main way used by most of people.When the growth degree of the nonlinear term is the critical exponent of Sobolev embedding, the embedding is continuous, but not compact, so the variation functional corresponding with the equation does not satisfy the compact conditions any more, which brings some difficulties in seeking the nontrivial solutions. In 1983, Brezis & Nireberg discussed the existence of positive solutions of semi-linear elliptic equation with critical exponent in bounded domains [1], soon after, in 1984, for unbounded domains, P.L.lions put forward a new principle called "concentration compactness principle" and applied it to treat a class of constrained variation problems in locally case [15]-[16], before long, in 1985, he put forward the second concentration compact principle for critical case as well. From then on, the Mountain Pass Lemma and the concentration-compactness method become the efficient method in concerning the existence of nontrivial solution of the nonlinear elliptic equation with critical exponent and the nonlinear elliptic equation in unbounded domains.In 1993, Zhang Guiyi considered the existence of the nontrivial solution of P-Laplace equation involving critical Sololev exponent in unbounded domains [17]. Since then, the discussions about the critical exponent developed rapidly. In 1998, Zhao Peihao studied the existence of two positive solutions of a class of semi-linear elliptic equations on bounded domains [3]. Professor Guo Xinkang developed this result, extend it to a class of quasi-linear elliptic equations on bounded. In this paper, we discussed the multiplicity of positive solutions of a class of quasi-linear elliptic equations with critical Sobolev exponent, and obtained two positive solutions of this equation in some conditions, extended the above result to a high level. This problem comes from the pervasion problems of physics, which possess application values in theory and practice.