Ground States for Singularly Perturbed Planar Choquard Equation with Critical Exponential Growth

Abstract

In this paper, we are dedicated to studying the following singularly Choquard equation $$−ε^2∆u + V (x)u = ε^{−α} [I_α ∗ F(u)] f(u), \ x ∈ \mathbb{R}^ 2,$$ where $V (x)$ is a continuous real function on $\mathbb{R}^2,$ $I_α : \mathbb{R}^2 → \mathbb{R}$ is the Riesz potential, and $F$ is the primitive function of nonlinearity $f$ which has critical exponential growth. Using the Trudinger-Moser inequality and some delicate estimates, we show that the above problem admits at least one semiclassical ground state solution, for $ε > 0$ small provided that $V (x)$ is periodic in $x$ or asymptotically linear as $|x| → ∞.$ In particular, a precise and fine lower bound of $\frac{f(t)}{e^{\beta_0 t^2}}$ near infinity is introduced in this paper.