Normalized ground states for Sobolev critical nonlinear Schrödinger equation in the <inline-formula><tex-math id="M1">$ L^2 $</tex-math></inline-formula>-supercritical case
Quanqing Li, Wenming Zou
Abstract
In this paper, we study the existence of the normalized ground state solutions to Sobolev critical nonlinear Schrödinger equation:$ \begin{array}{ll} \left\{ \begin{array}{ll} -\Delta u+\lambda u = f(u)+|u|^{2^*-2}u, \quad &{\hbox{in}}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2dx = m^2, \end{array} \right. \end{array}(P_m) $where $ N\geq 3 $, $ 2^*: = \frac{2N}{N-2} $, $ m>0 $, $ \lambda $ is unknown and will appear as a Lagrange multiplier, $ f $ is a mass supercritical and Sobolev subcritical nonlinearity. Using Pohozaev manifold and the concentration-compactness principle, we obtain a couple of the normalized solution to $ (P_m) $. The main contribution is related to the fact that we extend the results of L. Jeanjean, S. Lu published in 2020 on Calc. Var. [21] concerning the above problem from Sobolev subcritical setting to Sobolev critical setting, and our results answer an open problem raised by N. Soave published in 2020 on J. Funct. Anal. [37].