Study On Several Kinds Of Elliptic Equations

In this thesis we study several classes of elliptic equations and establish the existence,multiplicity and concentration of solutions.The thesis is organized as follows:In Chapter 1,we mainly give a introduction of the research background and current situa-tion.In Chapter 2,some basic knowledge is given to study these problems.In Chapter 3,we consider a problem of Schrodinger-Poisson involving concave-convex nonlinearities:where 1<q<2,4<p<6,the parameter A>0,the potential V = V+-V-,V? = ?V+-V-with V±= max {±V,0}.We assume the functions f,g,K,V satisfy suitable assumptions and obtain the existence and concentration of solutions via variational methods.In this chapter,we generalize the results of the semilinear elliptic problems in the existing references to the Schrodinger-Poisson equations.In the process of verifying the existence of the solutions,we define the corresponding Nehari manifold N?,divide N? into three parts N?+,N?0,N?-,and prove that under certain conditions,N?0= ?,N?±??.Moreover,the equations have different positive solutions on N?±;To verify the concentration of the solutions,we use Lions vanishing lemma and obtain that the limit of solutions are exactly the solutions of limit equations corresponding to the above Schrodinger-Poisson equations.The theory is significant to solve the Schrodinger-Poisson equations with concave-convex nonlinearity by using Nehari manifold.In Chapter 4,we consider the following fractional Kirchhoff problem where s?(0,1),N>2s,A>0 is a real parameter.We establish an existence result of positive ground states under some suitable conditions.A larg'e number of references have studied the fractional Kirchhoff equations with A.-R.conditions,in this chapter,we assume a condition which is weaker than A.-R.condition,and also obtain the existence of the ground states.As far as we know,the thought is innovative in the study of fractional equations.In Chapter 5,we consider a fractional quasilinear elliptic problem.where ?,?>0 are two parameters,with s ?(0,1)fixed,1<p<p<r<ps*,ps*? Np/(N-ps)the fractional Sobolev exponent and(=?)ps the fractional p-Laplacian operator.In the existing references,for the fractional p-Laplace equation,they usually assumed that the nonlinearity is superlinear,however,in this chapter we assume that the nonlinearity is a concave-convex nonlinearity.Since the sublinear term does not satisfy the A.-R.condition,the mountain pass theorem or other variational methods do not work.To overcome the above difficulties,in the process of verifying the multiplicity of solutions,we use the Brezis-Lieb lemma,the concentration-compactness lemma and Lusternik-Schnirelman theorem;.As to the concentra-tion of solutions,we mainly verify that the global maximum point of the solution concentrates at a local minimum point of the weight function.In Chapter 6,we summarize the content and innovation points of this thesis,and look ahead of the future research results.