Normalized solutions to nonlinear Schrödinger equations with competing Hartree‐type nonlinearities

Divyang Bhimani, Tianxiang Gou and Hichem Hajaiej

Mathematische Nachrichten, 2024, vol. 297, issue 7, 2543-2580

Abstract: In this paper, we consider solutions to the following nonlinear Schrödinger equation with competing Hartree‐type nonlinearities, −Δu+λu=|x|−γ1*|u|2u−|x|−γ2*|u|2uinRN,$$\begin{equation*} -\Delta u + \lambda u={\left(|x|^{-\gamma _1} \ast|u|^2\right)} u - {\left(|x|^{-\gamma _2} \ast|u|^2\right)} u\quad \mbox{in} \,\, \mathbb {R}^N, \end{equation*}$$under the L2$L^2$‐norm constraint ∫RN|u|2dx=c>0,$$\begin{equation*} \int _{\mathbb {R}^N}|u|^2 \, dx=c>0, \end{equation*}$$where N≥1$N \ge 1$, 0

Date: 2024
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