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Topological Methods in Nonlinear Analysis

Normalized solutions to a class of Choquard-type equations with potential
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Normalized solutions to a class of Choquard-type equations with potential

Authors

  • Lei Long
  • Xiaojing Feng

DOI:

https://doi.org/10.12775/TMNA.2023.028

Keywords

Choquard-type equations, normalized solutions, positive solutions

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 $a_0=0$.

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Published

2024-03-03

How to Cite

1.
LONG, Lei and FENG, Xiaojing. Normalized solutions to a class of Choquard-type equations with potential. Topological Methods in Nonlinear Analysis. Online. 3 March 2024. Vol. 63, no. 2, pp. 515 - 536. [Accessed 2 July 2025]. DOI 10.12775/TMNA.2023.028.
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Vol 63, No 2 (June 2024)

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Copyright (c) 2024 Lei Long, Xiaojing Feng

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This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

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