Ground States for Singularly Perturbed Planar Choquard Equation with Critical Exponential Growth (Journal of Nonlinear Modeling and Analysis, 2024)
In this paper, we are dedicated to studying the following singularlyChoquard 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 Rieszpotential, and $F$ is the primitive function of nonlinearity $f$ which has criticalexponential growth. Using the Trudinger-Moser inequality and some delicateestimates, we show that the above problem admits at least one semiclassicalground state solution, for $ε > 0$ small provided that $V (x)$ is periodic in $x$ orasymptotically linear as $|x| → ∞.$ In particular, a precise and fine lower boundof $\frac{f(t)}{e^{β_0 t^2}}$ near infinity is introduced in this paper.
Keywords: Choquard equation; critical exponential growth; Trudinger-Moser inequality; ground state solution.