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

Abstract

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

$$\begin{aligned} -\Delta u+V(x)u+\lambda u = |u|^{q-2}u+\beta |u|^{p-2}u, \quad \Vert u\Vert ^2_2=\int |u|^2dx = \alpha \end{aligned}$$

both on \({\mathbb R}^N\) as well as on domains \(r\Omega \) where \(\Omega \subset {\mathbb R}^N\) is a bounded smooth star-shaped domain and \(r>0\) is large. The exponents satisfy \(2<p<2+\frac{4}{N}<q<2^*=\frac{2N}{N-2}\), 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 \(\beta =0\).