Normalized solutions to Schödinger equations with potential and inhomogeneous nonlinearities on large convex domains

The paper addresses an open problem raised in [Bartsch, Molle, Rizzi, Verzini: Normalized solutions of mass supercritical Schrödinger equations with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence of normalized solutions to Schrödinger equations with potentials and inhomogeneous nonlinearities. We consider the problem $$-\mathrm{\Delta }u+V(x)u+\lambda u=\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\beta \mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,\mathrm{\parallel }u{\mathrm{\parallel }}_2^2=\int \mathrm{\mid }u{\mathrm{\mid }}^{2}dx=\alpha ,$$ both on ${\mathbb{R}}^N$ as well as on domains $r\mathrm{\Omega }$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is an open bounded convex domain and $r>0$ is large. The exponents satisfy $2<p<2+\frac{4}{N}<q<2^{\ast }=\frac{2N}{N-2}$, so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. 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 $\beta =0$.