Some Research On P-Laplacian Problem

The boundary value problem of differential equations involving the p-Laplacian operator is a kind of quasilinear elliptic differential equation used to describe diffusion phenomenon.It appears in mathematical theory and has a wide range of applications in various fields of science and technology,such as air flow and ocean motion,population model,nonlinear elasticity and fluid dynamics.Therefore,the study of this problem has theoretical value and practical significance due to its profound application background.This paper studies the existence of positive solutions for p-Laplacian problem,which is divided into five chapters:In chapter one,we briefly introduce the historical background and the research situation of the p-Laplacian problem,as well as the main work of this paper.In chapter two,we introduce some basic definitions and lemmas that will be used in this paper.In chapter three,we prove the existence of positive solutions for the p-Laplacian problem under superlinear conditions when the nonlinear term is nonnegative(that is,f(x,z)≥0).By studying the properties of the operator,the convexity of the solution and the continuity of the first eigenvalue,we transform the existence of solutions for the boundary value problems into the existence of fixed points of operators,and obtain new results.In chapter four,firstly,we extend the condition that f(x,z)≥0 to the case where f(x,0)≥0,and expand the nonlinear term.We obtain the existence of nonnegative and positive solutions by using the time-mapping.Then,we continue to weaken the limits of the nonlinear term,and consider the case that f(x,z)∈R.And the existence of positive solutions are obtained by using a maximum principle and the properties of the time-mapping.In chapter five,firstly,we summarize the main work and conclusions of this paper.And then point out the direction of further research about this topic in the future.In our research,the restriction on the nonlinear term f is gradually weakened,and the difficulties such as the lack of linearity of the operator are overcome.The existence results of positive solutions for the p-Laplacian problem in a wider range are obtained,which enrich the study of this problem.