Nonexistence Of Solutions For Several Nonlinear Elliptic Equations

In recent years,nonlinear elliptic partial differential equations have been widely used in many important fields such as natural science and engineering problems.The discussion on the existence and nonexistence of solutions of nonlinear elliptic partial differential equations attracts the attention of many scholars.This paper mainly discusses the nonexistence of solutions of several types of nonlinear partial differential equations.The main contents are given as follows:In the first chapter,we interview the background and developing trend of ?-Hessian equa-tions,a class of quasilinear elliptic equations with degenerate structure and briefly describe the main work and innovation point of this paper.In Chapter 2,we study the inclusion relationship of G(?)rding cone and ?-convex cone.In low dimensional cases,we learn relations between the two cones through simple examples directly.In higher dimensional case,we use the the nature of the cones and generalized examples to state and prove the inclusion relationship of the two cones.Finally,we discuss the operators defined in two cones contacted with ellipticity and admissibility.In Chapter 3,we study the nonexistence of positive entire solutions of the conformal ?-Hessian inequality.First,we recall the ?-Hessian Maclaurin inequality operator.Then we prove the main theorem by selecting proper test function and division of integral method.The proof is divided into two parts and uses Schwartz inequality,Young inequality.Finally,we give the example of existence for the solution of corresponding inequalities in second dimensional case.In Chapter 4,we study the nonexistence of the entire subsolutions of the ?-Hessian type equation.First,we introduce some properties of radial functions.Then,we state some related lemmas which play an important role in the proof of the main theorem.We also give some applications to the theorem with some special forms off,and discuss the existence of solutions by Keller-Osserman condition.In Chapter 5,we study the radial solutions of a quasilinear elliptic equation with a degen-erate structure.First,we discuss the Pohozaev identities of the quasilinear elliptic equation and then give Pohozaev-type identities of m(|x|)-Laplace equation and mean curvature equation.We also discuss the existence of solutions of the corresponding initial value problem and give the existence of solutions under certain conditions.In the last chapter,we summarize the research contents of this paper and give some prob-lems which could be studied in the future.