Study On The Existence And Multiplicity Of Solutions For P-Laplacian Elliptic Equations

In recent years,with the vigorous development of quantum physics,biology,economics and other applied disciplines,more and more scholars are interested in the study of p-Laplacian elliptic equations.In this paper,the variational method is used to study the existence and multiplicity of solutions for two kinds of p-Laplacian elliptic equations.By using the minimization principle,combined with Nehari manifolds and Faton Lemma,new results are obtained.Specifically,one is a p-biharmonic equation with concave-convex nonlinear term and a sign-changing weight function,and the other is a singular elliptic equation with Sobolev-Hardy critical exponent and sign-changing weight.These two kinds of equations come from physics and other disciplines,and have important application background and certain theoretical research value.The structure of this paper is as follows:In chapter 1,this chapter mainly introduces the research background,research status and related definitions of the above two kinds of equations.In chapter 2,we study the multiplicity of p-biharmonic equations with concave-convex nonlinear terms and sign-changing weight functions.Some restrictions on the nonlinear terms in the equation are given to ensure that the corresponding functional is meaningful in the appropriate space.under this condition,the variational method is used to prove the existence of multiple solutions in the equation.In chapter 3,we study the existence of solutions for singular elliptic equations with Sobolev-Hardy critical exponents with variable sign weights.Some assumptions are made for the nonlinear term in the equation so that the corresponding functional of the equation is forced,and then the variational method is used to prove that the equation has a positive solution.In chapter 4,it is a summary and prospect of the research results of this paper.