Normalized solutions for a class of nonlinear Choquard equations

Thomas Bartsch (), Yanyan Liu () and Zhaoli Liu ()
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Thomas Bartsch: Justus-Liebig-Universität Giessen
Yanyan Liu: Capital Normal University
Zhaoli Liu: Capital Normal University

Partial Differential Equations and Applications, 2020, vol. 1, issue 5, 1-25

Abstract: Abstract We prove the existence of a least energy solution to the problem $$\begin{aligned} -\Delta u-(I_{\alpha }*F(u))f(u)=\lambda u\ \text { in }\ {\mathbb {R}}^{N},\quad \int _{{\mathbb {R}}^N}u^2(x)dx = a^2, \end{aligned}$$ - Δ u - ( I α ∗ F ( u ) ) f ( u ) = λ u in R N , ∫ R N u 2 ( x ) d x = a 2 , where $$N\ge 1$$ N ≥ 1 , $$\alpha \in (0,N)$$ α ∈ ( 0 , N ) , $$F(s):=\int _{0}^{s}f(t)dt$$ F ( s ) : = ∫ 0 s f ( t ) d t , and $$I_{\alpha }:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$$ I α : R N → R is the Riesz potential. If f is odd in u then we prove the existence of infinitely many normalized solutions.

Keywords: Choquard equation; Stretched functional; Normalized solution; 35B38; 35J20; 35J60; 35P30 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s42985-020-00036-w

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