Study On The Solvability Of Two Types Of Elliptic Equations

This master’s thesis investigates two types of elliptic equations:the p-Laplace equation and the p-k-Hessian equation.For the p-Laplace equation problem,we investigate the existence of boundary blow-up solutions and boundary estimation by using the upper and lower solution method;for the p-k-Hessian equation,we study the existence of entire solutions by the monotone iterative method.We have divided the thesis into five chapters based on the content and the methods used,as follows:In Chapter 1,we focus on the study background,study purpose,study significance,and study status of boundary blow-up solutions to the p-Laplace equation and the entire solutions to the p-k-Hessian equation,and overview the content of this master thesis.In the last part of this chapter,we give the definitions and theorems that need to be used in the later part.In Chapter 2,we explore the existence of the boundary blow-up solutions and the boundary estimate to the p-Laplace equation.We provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplace equation.Moreover,when the weight function has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.In Chapter 3,we generalize the study content in Chapter 2.We study the existence of the boundary blow-up solutions and the asymptotic behavior of such solutions to the p-Laplace equation with gradient terms.We mainly study the sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem with gradient terms.In addition,we also analyze the existence of infinitely many radial solutions and nonexistence of boundary blow-up(radial)solutions when the weight function possesses strong singularity.In Chapter 4,we consider the existence of entire positive radial p-k-convex solutions for the p-k-Hessian equation and system.Our approach is based on a new monotone iteration scheme.Since the p-k-Hessian equation is the p-Laplace equation when k=1,the content of Chapter 4 improves and develops the content of Chapter 2.In Chapter 5,we summarise the conclusions of Chapter 2-4 of this master thesis,and describe the difficulties and innovations in each chapter.