TY - JOUR
T1 - Ground States for Singularly Perturbed Planar Choquard Equation with Critical Exponential Growth
AU - Zhang , Limin
AU - Liao , Fangfang
AU - Tang , Xianhua
AU - Qin , Dongdong
JO - Journal of Nonlinear Modeling and Analysis
VL - 2
SP - 247
EP - 271
PY - 2023
DA - 2023/08
SN - 5
DO - http://doi.org/10.12150/jnma.2023.247
UR - https://global-sci.org/intro/article_detail/jnma/21924.html
KW - Choquard equation, critical exponential growth, Trudinger-Moser
inequality, ground state solution.
AB - <p style="text-align: justify;">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 $ε &gt; 0$ 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.</p>