On The Solutions To Elliptic Equations And Systems Involving Singular Potentials And Critical Exponents

In this paper,the main research object is several classes biharmonic equation(systems)with Rellich potentials and the elliptic equations with Hardy potentials,which is divided into four chapters.In the first chapter,mainly introduced the problems,the background and main conclusions.In the second chapter,we mainly study the biharmonic equation with Rellich potentials,firstly by using variable substitution and Kelvin transform to obtain limit equation of ground state solutions of asymptotic behavior,which results in the second and third chapter in order to establish the corresponding equation(systems)of energy functional of local(PS)condition play a very important role;Finally,the existence of solutions for a class of biharmonic equations with linear perturbation is proved.In Chapter 3,we first use the results in Chapter 2 to study the best constants of a class of Rellich-Sobolev.Furthermore,the existence of nontrivial solutions of a system with multiple Rellich terms and strong coupled critical terms is proved by using this optimal constants to establish the required(PS)conditions.In the fourth chapter,we mainly study a class of elliptic equations with Hardy and strongly coupled critical terms.First,we discuss the asymptotic behavior of the ground state solution of the system by using ordinary differential equation analysis method.Furthermore,the local(PS)condition for a class of equations with linear perturbation is established,and the existence of the Mountain-path-type solution is proved.