Existence and concentration of ground states to a quasilinear problem with competing potentials

Abstract

In this paper, we are concerned with the existence and concentration behavior of ground states for the following quasilinear problem with competing potentials −ε2Δu+V(x)u−ε212Δ(u2)u=P(x)|u|p−1u+Q(x)|u|q−1u, where 3<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 ε→0+.