Liejun Shen, Marco Squassina
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 4, 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.
DOI: 10.58997/ejde.2025.34
Show me the PDF file (497 KB), TEX file for this article.
| Liejun Shen Department of Mathematics Zhejiang Normal University Jinhua, Zhejiang 321004, China email: ljshen@zjnu.edu.cn | |
 |
Marco Squassina Dipartimento di Matematica e Fisica Università Cattolica del Sacro Cuore Via della Garzetta 48, 25133, Brescia, Italy email: marco.squassina@unicatt.it |
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