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 \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda u+h(x)|u|^{q-2}u+|u|^{2^*-2}u&{}\text{ in }\ {\mathbb {R}}^N, \\ \int _{{\mathbb {R}}^N}|u|^2\textrm{d}x=a,\\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u=\lambda u+{h(x)|u|}^{q-2}u+{|u|}^{2^{\ast }-2}u\hfill & \text{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^2\text{d}x=a,\hfill \end{array}\right)\end{array}$$ where \(N\ge 3\) $N\ge 3$ , \(a>0\) $a>0$ , \(\lambda \in {\mathbb {R}} \) $\lambda \in \mathbb{R}$ , \(h\ne const\) $h\ne const$ and \(2^*=\frac{2N}{N-2}\) $2^{\ast }=\frac{2N}{N-2}$ is the Sobolev critical exponent. In the \(L^2\) $L^2$ -subcritical regime (i.e. \(2<q<2+\frac{4}{N}\) $2<q<2+\frac{4}{N}$ ), 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 \(L^2\) $L^2$ -subcritical regime, it is necessary to apply some reverse conditions to h provided that at least \(L^2\) $L^2$ -critical regime (i.e. \(2+\frac{4}{N}\le q<2^*\) $2+\frac{4}{N}\le q<2^{\ast }$ ) 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.