Normalized solutions to Schrödinger equations with potential and inhomogeneous nonlinearities on large smooth domains

The paper addresses an open problem raised in Bartsch et al. (Commun Partial Differ Equ 46(9):1729–1756, 2021) on the existence of normalized solutions to Schrödinger equations with potentials and inhomogeneous nonlinearities. We consider the problem \(\begin{aligned} -\Delta u+V(x)u+\lambda u = |u|^{q-2}u+\beta |u|^{p-2}u, \quad \Vert u\Vert ^2_2=\int |u|^2dx = \alpha \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+V(x)u+\lambda u={|u|}^{q-2}u+{\beta |u|}^{p-2}{u,\Vert u\Vert }_2^2=\int {|u|}^{2}dx=\alpha \end{array}$$ both on \({\mathbb R}^N\) ${\mathbb{R}}^N$ as well as on domains \(r\Omega \) $r\mathrm{\Omega }$ where \(\Omega \subset {\mathbb R}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded smooth star-shaped domain and \(r>0\) $r>0$ is large. The exponents satisfy \(2<p<2+\frac{4}{N}<q<2^*=\frac{2N}{N-2}\) $2<p<2+\frac{4}{N}<q<2^{\ast }=\frac{2N}{N-2}$ , so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. Nonlinear Schrödinger equations with combined power-type nonlinearities have been investigated first by Tao et al. (Commun Partial Differ Equ 32(7-9):1281-1343, 2007). Due to the presence of the potential a by now standard approach based on the Pohozaev identity cannot be used. We develop a robust method to study the existence of normalized solutions of nonlinear Schrödinger equations with potential and find conditions on V so that normalized solutions exist. Our results are new even in the case \(\beta =0\) $\beta =0$ .