Existence And Concentration Of Ground States To Quasilinear Problems

In this paper, we first study following quasilinear Schrodinger equation with competing potentials where3<q<p<22*-1,2*is the Sobolev critical exponent, V(x) and P(x) are positive and Q(x) may be sign-changing. We show the existence of the ground states via the Nehari manifold method for ε>0, and these ground states "concentrate" at a global minimum point of the least energy function C(s) as εâ†'0.Furthermore, we consider the following sub-cubic problemwhere1<p<3, V(x) has global minimum and K(x) has global maximum, by minimizing the energy functional on Pohozaev manifold, we show the existence of ground states for ε>0. Furthermore, we found that these ground states converges to a ground stete solution of the associated limit problem and concentrates to some set related to the linaer potential V and nonlinear potential K.