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: \[\{{aligned} & -Δ u+λ u=t|u|q-2u+|u|^{2*-2}u RN,\\ & u∈ H¹(RN), {aligned}.\] where N≥ 3 , $t>0$ , λ >0 and 2< q<2*=2N/N-2 . Based on our recent study on the normalized solutions of the above equation 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].], we prove that (1) the above equation has two positive radial solutions for $N=3$ , $2< q<4$ and $t>0$ 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 tq*>0 for 2< q≤ 4 such that the above equation has ground-states for t≥ tq* in the case of $2< q<4$ and for t>t₄* in the case of $q=4$ , while the above equation has no ground-states for 0< t< tq* for all 2< q≤ 4 , which, together with the well-known results on ground-states of the above equation, almost completely solve the existence of ground-states, except for $N=3$ , $q=4$ and t=t₄* . Moreover, based on the almost completed study on ground-states to the above equation, we introduce a new argument to study the normalized solutions of the above equation to prove that there exists 0<t̄a,q<+∞ for 2< q<2+4/N such that the above equation has no positive normalized solutions for t>t̄a,q with ∫ _RN|u|² dx=a² , 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: \[\{ {aligned} & -Δ u+λ u+(x_1²+x_2²)u=|u|ᵖ⁻²u R³,\\ & u∈ H¹(R³), ∫_{R³}|u|²{ d}x=r², {aligned}.\] where x=(x₁,x₂,x₃)∈ R³ , 10/3< p<6 , $r>0$ is a constant and (u, λ ) is a pair of unknowns with λ being