Multiple Solutions For Nonlinear Elliptic Equations With The Right Problem

In this thesis, by applying variational or non-variational methods, such as critical point theory (smooth or nonsmooth), fixed point theorem, we discuss the existence and multiplicity of solutions for several kinds of elliptic boundary value problems. This thesis is composed by three parts, the main contents are as follows:In Part 1, we discuss a class of elliptic boundary value problems involving diver-gence type operator where p> N≥2,αand g are positive radial continuous functions. Using the minimax principle, we first obtain the existence of non-radial ground state solutions for the above problem, and then by applying the Ljusternik-Schnirelmann category theorems, we get the existence of multiple solutions for the problems.In Parts 2, we study two kinds of boundary value problems involving singular and nonsmooth nonlinearities and where is measurable and locally Lipschitz in the u-variable,(?) F(x, u) denotes the generalized gradient of the function u→F(x, u) (hence (?)F(x, u) may not be continuous)and f is a measurable function (probably not continuous). By applying nonsmooth critical point theory, under some conditions, we obtain the existence of at least two solutions for both above boundary value problems.We point out that our methods dealing with boundary value problems (smooth or nonsmooth) in above parts are variational, in Part 3, by apply a fixed point theorem, we show that the existence of solutions for the following elliptic equation on unbounded domain RN involving singular and nonsmooth nonlinearities: Multiple solutions for elliptic equations with weights Abstract where N≥3, H is the Heaviside function, namely...