Normalized solutions of the Schrödinger equation with potential

Xin Zhao and Wenming Zou

Mathematische Nachrichten, 2024, vol. 297, issue 5, 1632-1651

Abstract: In this paper, for dimension N≥2$N\ge 2$ and prescribed mass m>0$m>0$, we consider the following nonlinear scalar field equation with L2$L^2$ constraint: −Δu+V(x)u+λu=g(u)inRN,∫RNu2=m,$$\begin{equation*} \left\{ \def\eqcellsep{&}\begin{array}{l} -\Delta u+V(x)u+\lambda u=g(u)\qquad \hbox{in} \; \mathbb {R}^N, \\ \int _{\mathbb {R}^N} u^2=m, \end{array} \right. \end{equation*}$$where λ∈R$\lambda \in \mathbb {R}$ is a Lagrange multiplier, V(x)∈C1(RN,R)$V(x)\in C^1 (\mathbb {R}^N,\mathbb {R})$. In particular, g(x)∈C(R,R)$g(x)\in C(\mathbb {R},\mathbb {R})$ satisfies mass supercritical and Sobolev subcritical growth. We prove the existence results of the normalized solution and infinitely many normalized solutions to the above system under some proper assumptions on the functions V(x),g(x)$V(x), g(x)$ by the mountain pass argument.

Date: 2024
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https://doi.org/10.1002/mana.202100498

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