Semiclassical ground state solutions for a Choquard type equation in ${ℝ}^2$ with critical exponential growth

Yang, Minbo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 177-209.

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

$$-{\epsilon }^2\Delta u+V\left(x\right)u{=}^{\mu -2}\left[\frac{1}{{\left|x\right|}^{\mu }}*\left(P\left(x\right)G\left(u\right)\right)\right]P\left(x\right)g\left(u\right)$$

in ${ℝ}^2$, where $\epsilon$ is a positive parameter, $\frac{1}{{\left|x\right|}^{\mu }}$ with $0<\mu <2$ is the Riesz potential, $*$ is the convolution operator, $V\left(x\right)$, $P\left(x\right)$ are two continuous real functions and $G\left(s\right)$ is the primitive function of $g\left(s\right)$. Suppose that the nonlinearity $g$ is of critical exponential growth in ${ℝ}^2$ 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 $P\left(x\right)$ if $V\left(x\right)$ is a constant.

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