Semiclassical ground state solutions for a Choquard type equation in ℝ² with critical exponential growth^∗

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, P.R. China.
mbyang@zjnu.edu.cn

Received: 28 September 2016
Accepted: 25 January 2017

Abstract

In this paper we study a nonlocal singularly perturbed Choquard type equation

$\mathrm{-}{\mathit{\epsilon }}^{\mathrm{2}}\mathrm{\Delta }\mathit{u}\mathrm{+}\mathit{V}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{u}\mathrm{=}{\mathit{\epsilon }}^{\mathit{\mu }\mathrm{-}\mathrm{2}}\left[\frac{\mathrm{1}}{\mathrm{|}\mathit{x}{\mathrm{|}}^{\mathit{\mu }}}\mathrm{\ast }(\mathit{P}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{G}\mathrm{\left(}\mathit{u}\mathrm{\right)})\right]\mathit{P}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{g}\mathrm{\left(}\mathit{u}\mathrm{\right)}$ in ℝ², where ε is a positive parameter, $\frac{\mathrm{1}}{\mathrm{|}\mathit{x}{\mathrm{|}}^{\mathit{\mu }}}$ with 0 < μ < 2 is the Riesz potential, ∗ is the convolution operator, V(x), P(x) are two continuous real functions and G(s) is the primitive function of g(s). Suppose that the nonlinearity g is of critical exponential growth in ℝ² in the sense of the Trudinger-Moser inequality, we establish some existence and concentration results of the semiclassical solutions of the Choquard type equation in the whole plane. As a particular case, the concentration appears at the maximum point set of P(x) if V(x) is a constant.