Existence And Multiplicity Of Solutions For A Class Of Quasilinear Elliptic Equations With Critical Growth

In this paper,we consider a class of quasilinear elliptic equations with Sobolev exponent-?p?pu-?p?p(u2)u + V(x)|u|p-2u = h(u)+ K(x)|u|2p*-2u,x ? RN,(1)where ?pu = div(|(?)u|p-2(?)u)is the usual p-Laplacian,2?p<N,p*= Np/N-p is Sobolev critical exponent.The reduction form of equation(1)arises from the physical model-s such as fluid mechanics,plasma physics,Heidelberg ferromagnetism and condensed matter theory.Under suitable assumptions on the potential V(x),K(x)and the non-linear term h(u),we will prove the existence,multiple and concentration behavior of solutions for equation(1).The paper will be divided into four chapters as follows:Chapter 1 We introduce some research backgrounds,recent research progress and state the content of this paper.Chapter 2 We state Sobolev space,inequalities and lemmas which will be used in the sequel.Chapter 3 When ? = 1,V(x)= 1,h(u)= 0,we will apply the constrained mini-mizing methods and concentration-compactness at infinity to establish the existence of nontrival solutions for equation(1)under K(x)satisfies suitable assumptions.Chapter 4 When K(x)? 1,we will use the Ljusternik-Schnirelman category,Nehari manifold and minimax methods to prove the existence of the ground state solu-tions,multiple solutions and concentration behavior of solutions for equation(1)under the boundedness assumption V(x)and the subcritical assumption of h(u).