On some nonlinear Schrödinger equations in ℝN

In this paper, we consider the following nonlinear Schrödinger equations with the critical Sobolev exponent and mixed nonlinearities: \[\left\{\begin{aligned} & -\Delta u+\lambda u=t|u|^{q-2}u+|u|^{2^{*}-2}u\quad\text{in }\mathbb{R}^{N},\\ & u\in H^{1}(\mathbb{R}^{N}), \end{aligned}\right.\] where $N\geq 3$, $t>0$, $\lambda >0$ and $20$ sufficiently large, which gives a rigorous proof of the numerical conjecture in [J. Dávila, M. del Pino and I. Guerra. Non-uniqueness of positive ground states of non-linear Schrödinger equations. Proc. Lond. Math. Soc. 106 (2013), 318–344.]; (2) there exists $t_q^{*}>0$ for $2t_4^{*}$ in the case of $q=4$, while the above equation has no ground-states for $0\overline {t}_{a,q}$ with $\int _{\mathbb {R}^{N}}|u|^{2}{\rm d}x=a^{2}$, which, together with our recent study in [J. Wei and Y. Wu, Normalized solutions for Schrodinger equations with critical Sobolev exponent and mixed nonlinearities, e-print arXiv:2102.04030[Math.AP].], gives a completed answer to the open question proposed by Soave in [N. Soave. Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case. J. Funct. Anal. 279 (2020) 108610.]. Finally, as applications of our new argument, we also study the following Schrödinger equation with a partial confinement: \[\left\{ \begin{aligned} & -\Delta u+\lambda u+(x_1^{2}+x_2^{2})u=|u|^{p-2}u\quad\text{in }\mathbb{R}^{3},\\ & u\in H^{1}(\mathbb{R}^{3}),\quad \int_{\mathbb{R}^{3}}|u|^{2}{\rm d}x=r^{2}, \end{aligned}\right.\] where $x=(x_1,x_2,x_3)\in \mathbb {R}^{3}$, $\frac {10}{3}0$ is a constant and $(u, \lambda )$ is a pair of unknowns with $\lambda$ being a Lagrange multiplier. We prove that the above equation has a second positive solution, which is also a mountain-pass solution, for $r>0$ sufficiently small. This gives a positive answer to the open question proposed by Bellazzini et al. in [J. Bellazzini, N. Boussaid, L. Jeanjean and N. Visciglia. Existence and Stability of Standing Waves for Supercritical NLS with a Partial Confinement. Commun. Math. Phys. 353 (2017), 229–251].