Several Kinds Of Semi-positone Elliptic Equations And Related Problems With Dirichlet Boundry Conditions

In recent years,the boundary value problem of semipositone elliptic equations has attracted increasing attention from scholars at home and abroad,which can describe and solve many natural phenomena and engineering and technical problems in our real life.Especially for applications in mechanical systems,suspension bridge design,astro-physics,combustion theoretical models,etc.At present,there are many achievements in the study of boundary value problems for semipositone elliptic equations.In this paper,we discuss the existence of solutions for two classes of semipositone elliptic equations with Dirichlet boundary conditions.Firstly,the existence of radial positive solutions for a class of semipositone elliptic equations is studied.We first use the inverse method and with the help of the properties of the first eigenvalue ?₁the corresponding eigenfunction of the -? with Dirichlet boundary condition in the?,it is proved that when the ? is sufficient,the equation has no non-negative solution.Secondly,the variational method theory is used to prove that the equation has a positive radial solution when the ? is sufficiently large,and it is proved that the linearization operator at each solution has a nonnegative first eigenvalue,so the solution is stable.In the second part,we study the existence of positive solutions for a class of semipositone elliptic equations under three different hypothetical conditions.We first study that f is sublinear at infinity,here we discuss two cases,and then we discuss the singular case of f at origin.These theorems gave a generation to Kaufmann and Quoirin's work.