Existence and concentration of ground state solutions for a quasilinear Choquard equation with critical exponential growth in ℝ2

Abstract

In this paper, we consider the following quasilinear Choquard equation − ϵ 2 Δ u + V ( x ) u − ϵ 2 Δ ( u 2 ) u = ϵ μ − 2 ( 1 | x | μ ∗ F ( u ) ) f ( u ) in R 2 , where ϵ > 0 is a parameter, 0 < μ < 2 , ∗ is the convolution product in R 2 , V ( x ) is a continuous real function in R 2 , F ( u ) is the primitive function of f ( u ) and f has critical exponential growth with respect to the Trudinger-Moser inequality. By employing a change of variables, the quasilinear equation can be reduced to a semilinear equation, whose associated functional is well defined in a nonstandard Orlicz space and exhibits a mountain pass geometry. Under suitable assumptions on V and f, we investigate the existence and concentration behavior of positive ground state solutions for the above problem by variational methods.