Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities

Abstract

We study the Sobolev critical Schrödinger equation with combined power nonlinearities

$$\begin{aligned} -\Delta u=\lambda u+|u|^{\frac{2N}{N-2}-2}u+\mu |u|^{q-2}u,\ x\in {\mathbb {R}}^{N} \end{aligned}$$

having prescribed mass

$$\begin{aligned} \int _{{\mathbb {R}}^N}|u|^2dx=a^2. \end{aligned}$$

For a $L^2$-critical or $L^2$-supercritical perturbation $\mu |u|^{q-2}u$, we prove existence of normalized ground states, by introducing the Sobolev subcritical approximation method to mass constrained problem. Our result settles a question raised by N. Soave [22]. Meanwhile, the Sobolev subcritical problem is treated again by using the Pohožaev constraint, Schwartz symmetrization rearrangements and various scaling transformations.