Existence Of Ground State Solutions And Infinite Solutions For A Class Of Quasilinear Elliptic Equations In R~N

In this paper, we consider a class of the following quasilinear elliptic equations: where u(x):IRNâ†'IR, â–³pu= div(|â–½p-2â–½u) is the usual p-Laplacian. Under some appropriate assumptions on the potential function V(x) and the nonlinear term g(u), we will prove the existence of ground state solutions and infinite solutions for equation (1). The paper is organised as follows:Chapter 1 We introduce some research backgrounds and research status about equation (1). Meanwhile, we list some space and several inequalities which will be used later.Chapter 2 Assume that V(x) is bounded and g(u)=|u|r-1u, where 1< p< N, p-1< r<2p*-1 (p*=Np/N-p), we apply the minimization method under Pohozaev constraint to establish the existence of ground state solutions for equation (1).Chapter 3 Assume that V(x) is positive lower bounded and g(u) has super-p growth, where 2≤p< N, we use the Fountain theorem and the change of variables method to prove the existence of infinite solutions for equation (1).