Normalized Solutions of Non-autonomous Schrödinger Equations Involving Sobolev Critical Exponent

In this paper, we look for normalized solutions to the following non-autonomous Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda u+h(x)|u|^{q-2}u+|u|^{2^*-2}u&{}\text{ in }\ {\mathbb {R}}^N, \\ \int _{{\mathbb {R}}^N}|u|^2\textrm{d}x=a,\\ \end{array} \right. \end{aligned}$$

where \(N\ge 3\), \(a>0\), \(\lambda \in {\mathbb {R}} \), \(h\ne const\) and \(2^*=\frac{2N}{N-2}\) is the Sobolev critical exponent. In the \(L^2\)-subcritical regime (i.e. \(2<q<2+\frac{4}{N}\)), by proposing some new conditions on h, we verify that the corresponding Pohozaev manifold is a natural constraint and establish the existence of normalized ground states. Compared to the \(L^2\)-subcritical regime, it is necessary to apply some reverse conditions to h provided that at least \(L^2\)-critical regime (i.e. \(2+\frac{4}{N}\le q<2^*\)) is considered. We prove the existence of minimizer on the Pohozaev manifold of the associated energy functional and determine that the minimizer is a normalized solution by using the classical deformation lemma. In particular, by imposing further assumptions on h, the ground states can be obtained.