  The paper addresses an open problem raised in [Bartsch, Molle, Rizzi,
Verzini: Normalized solutions of mass supercritical Schr\"odinger equations
with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence
of normalized solutions to Schr\"odinger equations with potentials and
inhomogeneous nonlinearities. We consider the problem \[
  -\Delta u+V(x)u+\lambda u = |u|^{q-2}u+\beta |u|^{p-2}u, \quad
\|u\|^2_2=\int|u|^2dx = \alpha, \] 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 exponents satisfy $2<p<2+\frac4N<q<2^*=\frac{2N}{N-2}$,
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
Schr\"odinger equations with potential and find conditions on $V$ so that
normalized solutions exist. Our results are new even in the case $\beta=0$.
