Normalized solutions for Schr\"odinger equations with critical Sobolev exponent and mixed nonlinearities

TL;DR

This paper investigates the existence, nonexistence, and asymptotic behavior of solutions to nonlinear Schr"odinger equations with mixed nonlinearities and critical Sobolev exponent, providing new insights into solution structures under various parameter regimes.

Contribution

It establishes existence and nonexistence results for solutions with mixed nonlinearities, and describes their asymptotic behaviors as parameters vary, addressing open questions in the field.

Findings

Existence of mountain-pass solutions for N=3 and 2<q<2+4/N.

Existence and nonexistence of ground states for certain q ranges with large .

Asymptotic behaviors of solutions as  approaches zero or its upper bound.

Abstract

In this paper, we consider the following nonlinear Schr\"{o}dinger equations with mixed nonlinearities: \begin{eqnarray*} \left\{\aligned &-\Delta u=\lambda u+\mu |u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ &u\in H^1(\bbr^N),\quad\int_{\bbr^N}u^2=a^2, \endaligned\right. \end{eqnarray*} where $N\geq3$, $\mu>0$, $\lambda\in\mathbb{R}$ and $2<q<2^*$. We prove in this paper \begin{enumerate} \item[$(1)$]\quad Existence of solutions of mountain-pass type for $N=3$ and $2<q<2+\frac{4}{N} $. \item[$(2)$]\quad Existence and nonexistence of ground states for $2+\frac{4}{N}\leq q<2^*$ with $\mu>0$ large. \item[$(3)$]\quad Precisely asymptotic behaviors of ground states and mountain-pass solutions as $\mu\to0$ and $\mu$ goes to its upper bound. \end{enumerate} Our studies answer some questions proposed by Soave in \cite[Remarks~1.1, 1.2 and 8.1]{S20}.