Multiplicity of normalized solutions for a Schrödinger equation with critical growth in ${\mathbb{R}}^N$

Abstract

In this paper we study the multiplicity of normalized solutions to the following nonlinear Schrödinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-Δu=λu+μ|u|^{q-2}u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a,\mu >0$, $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $q\in (2,2+\frac{4}{N})$ and $f$ has an exponential critical growth when $N=2$, and $f(u)=|u{|}^{2^{\ast }-2}u$ when $N\ge 3$ and $2^{\ast }=\frac{2N}{N-2}$.

arXiv admin note: text overlap with arXiv:2102.03001. text overlap with arXiv:1811.04044 by other authors