Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 34, pp. 1-23.

Title: Concentrating normalized solutions for 2D nonlocal Schrodinger equations with critical exponential growth

Authors: Liejun Shen (Zhejiang Normal Univ., Jinhua, Zhejiang, China)
         Marco Squassina (Univ. Cattolica del Sacro Cuore, Brescia, Italy)

Abstract:
We study the existence of solutions to nonlocal Schrodinger problems
with different types of potentials
$$\displaylines{
-\Delta u +W(x) u=\sigma u +\kappa[|x|^{-\mu}\ast F(u)]f(u)\quad
\text{in }\mathbb{R}^2, \cr
\int_{\mathbb{R}^2}|u|^2dx=a^2,
}$$
where $a\neq0$, $\sigma\in\mathbb{R}$ is known as the Lagrange multiplier, 
$\kappa>0$ is a parameter, $W\in \mathcal{C}(\mathbb{R}^2)$ is
the nonnegative external potential, $\mu\in(0,2)$, 
and $F$ denotes the primitive function of $f\in \mathcal{C}(\mathbb{R})$ which 
has critical exponential growth in the Trudinger-Moser sense at infinity.
We prove that the problems admit at least a positive solution,
and we analyze the concentrating behavior.

Submitted January 12, 2025. Published April 04, 2025.
Math Subject Classifications: 35A15, 35J10, 35B09, 35B33.
Key Words: Positive normalized solution; Choquard equation,
 critical exponential growth;  Rabinowitz's type potential;
 steep potential well; variational method.