A Class Of Elliptic Equations And The Existence Of Solutions Of Elliptic Systems

The variational method is an approximate calculation method which based on theariational principle. It is one of the important methods in computational mechanics,nd an efective tool to solve the problem of mechanics and other areas. The studybject of Variational is extremum problem of the functional. The variational princi-le is a variational form representations of physical laws, and it does not contain newhysical content. Its value is mainly used as the new starting point for the structuref classical mechanics theory. In other words, in all motion state of satisfy certain con-traints, actual motion state should make some physical quantity the extreme value.n mechanics, optics, quantum mechanics and other subjects, it has the correspondingariational principle. There are some important variational principles such as Fermat’srinciple, Hamiltonian principle, the principle of minimum energy and minimum resid-al energy principle. The variational method is approximation calculation method inhe field of mechanics, physics and other disciplines.In this paper, the deformation of the fountain theorem and fountain theorem, wetudied with superlinear and asymptotically linear elliptic equations and superquadraiclliptic systems and give some sufcient conditions of existences of solutions. Meanwhilee apply the main results to the existence of solutions for boundary value problems.The thesis is divided into three chapters according to contents.In chapter1, it introduce the origin and setting of variational method.In chapter2, based on a variant fountain theorem established by W. Zou, we studyhe following elliptic boundary value problemFor superlinear case, in sharp contrast to the existing results in the literature, we doot make the well-known Ambrosetti-Rabinowitz type technical condition near infinitynd not make any assumptions near zero on the behaviors of the nonlinearity f, webtain the existence and multiplicity of solutions. We also consider the situation where is asymptotically linear with assumptions diferent from those studied previously, and In chapter3, we study the existence of solution for the following elliptic system...