Existence Of Solutions For Some Quasilinear Elliptic Equations

In recent decades, the research on partial differential equations is very active. The study of this field has important theoretical guidance and practical significance to the development of mathematics and other subjects. The quasilinear elliptic equation can be seen as a very important class of partial differential equations. The p-Laplace equation from the fluid and the porous medium equation are the typical example of quasilinear elliptic equation. In this thesis, we mainly study the following aspects of some important quasilinear elliptic equations.In chapter one, the background and significance of the research are introduced. And the current research process at home and abroad in this direction are reviewed. In addition, we outline the four kinds of equations are studied in this thesis and the appropriate assumptions for the equations.In chapter two, we list some bascic inequalities, theorems and propositions related to our thesis.In chapter three, first, the definitions of weak solutions for the above equations are given. Then, we mainly study the existence of weak solutions for three kinds of quasilinear p-Laplace equations with L∞datum(that is a(x), f(x), B(x, u, u) L()∞? ∈ ?) in bounded domain by using Schauder’s fixed point theorem and variational method. Finally, by virtue of approximation technique and energy method, we prove the existence of weak solutions for three kinds of quasilinear p-Laplace equations with1 L datum(that is 1a(x), f(x), B(x, u, ?u) ∈L(?)) in bounded domain and the existence of nonnegative weak solutions for a class of non-standard singular elliptic equations with nonnegative measure in the right-hand side.