Mengfei Tao, Binlin Zhang, Normalized solutions for a nonlinear Schrödinger equation via a fixed point theorem

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DOI: 10.23952/jnva.9.2025.3.03

Volume 9, Issue 3, 1 June 2025, Pages 357-371

Abstract. In this paper, we aim to investigate the existence and boundedness of the normalized solutions of the following nonlinear Schrödinger equation by using a non-variational fixed point theorem:
$- \Delta u + V(x)u = \lambda u + f(u) + h$ in $\Omega,$
$\int_{\Omega} |u|^{2} dx = c$
$u = 0$ on $\partial \Omega,$
where the potential function $V(x):\Omega \rightarrow R$ is measurable and can change sign, $h\not \equiv 0$ is the perturbation term, the nonlinearity $f$ is measurable and satisfies critical or subcritical growth for $N \geq 3$ , exponential critical growth for $N = 2$ . We first consider $\Omega$ as the whole space $R^N$ and establish the existence of normalized solutions to the problem for $N=2$ and $N \geq 3$ , respectively. While $\Omega$ is considered as a bounded domain, we establish the existence and $L^{\infty}$ -boundedness of the normalized solution of the problem for $N \geq 3$ , where the nonlinearity $f$ satisfies the subcritical growth condition. In the bounded domain $\Omega$ , we can still obtain the existence of normalized solutions to the above problem with the critical growth condition instead of the $L^{\infty}$ -boundedness of normalized solutions. As far as we know, our results are more general and novel on this topic, and can also provide a new approach for the study of nonlinear Schrödinger equation with prescribed $L^2$ -norm.

How to Cite this Article:
M. Tao, B. Zhang, Normalized solutions for a nonlinear Schrödinger equation via a fixed point theorem, J. Nonlinear Var. Anal. 9 (2025), 357-371.