Normalized solutions for Schödinger equations with potential and general nonlinearities involving critical case on large convex domains

Abstract

In this paper, we study the following Schrödinger equations with potentials and general nonlinearities \begin{equation*} \left\{\begin{aligned} & -Δu+V(x)u+λu=|u|^{q-2}u+βf(u), \\ & \int |u|^2dx=Θ, \end{aligned} \right. \end{equation*} both on ${\mathbb{R}}^N$ as well as on domains $r\Omega$ where $\Omega \subset {\mathbb{R}}^N$ is an open bounded convex domain and $r>0$ is large. The exponent satisfies $2+\frac{4}{N}\le q\le 2^{\ast }=\frac{2N}{N-2}$ and $f:\mathbb{R}\to \mathbb{R}$ satisfies $L^2$-subcritical or $L^2$-critical growth. This paper generalizes the conclusion of Bartsch et al. in \cite{TBAQ2023}(2023, arXiv preprint). Moreover, we consider the Sobolev critical case and $L^2$-critical case of the above problem.

58pages. arXiv admin note: substantial text overlap with arXiv:2306.07826 by other authors