2024 Normalized solutions to a class of Choquard-type equations with potential
Lei Long, Xiaojing Feng
Topol. Methods Nonlinear Anal. 63(2): 515-536 (2024). DOI: 10.12775/TMNA.2023.028

Abstract

In this paper, we study the existence and nonexistence of solutions to the following Choquard-type equation\begin{equation*}-\Delta u+(V+\lambda)u=(I_\alpha*F(u))f(u)\quad\text{in } \mathbb{R}^N,\end{equation*}having prescribed mass $\int_{\mathbb{R}^N}u^2=a$, where $\lambda\in\mathbb{R}$ will arise as a Lagrange multiplier, $N\geq 3$, $\alpha\in(0,N)$, $I_\alpha$ is Riesz potential. Under suitable assumptions on the potential function $V$ and the nonlinear term $f$, $a_0\in[0,\infty)$ exists such that the above equation has a positive ground state normalized solution if $a\in(a_0,\infty)$ and one has no ground state normalized solution if $a\in(0,a_0)$ when $a_0> 0$ by comparison arguments. Moreover, we obtain sufficient conditions for $a_0=0$.

Citation

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Lei Long. Xiaojing Feng. "Normalized solutions to a class of Choquard-type equations with potential." Topol. Methods Nonlinear Anal. 63 (2) 515 - 536, 2024. https://doi.org/10.12775/TMNA.2023.028

Information

Published: 2024
First available in Project Euclid: 17 July 2024

Digital Object Identifier: 10.12775/TMNA.2023.028

Keywords: Choquard-type equations , normalized solutions , ‎positive‎ ‎solutions

Rights: Copyright © 2024 Juliusz P. Schauder Centre for Nonlinear Studies

Vol.63 • No. 2 • 2024
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