J. Nonl. Mod. Anal., 5 (2023), pp. 247-271.

Published online: 2023-08

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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.

}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2023.247}, url = {http://global-sci.org/intro/article_detail/jnma/21924.html} }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.

*Journal of Nonlinear Modeling and Analysis*.

*5*(2). 247-271. doi:10.12150/jnma.2023.247