Normalized solutions of mass subcritical Schrödinger equations in exterior domains

Abstract

In this paper, we study the nonlinear Schrödinger equation with $L^2$-norm constraint:

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta u=\lambda u+|u|^{p-2}u\quad &{}\text {in}\ ~~\Omega ,\\ u= 0 &{}\text {on}\ ~\partial \Omega ,\\ \int _{\Omega }|u|^2dx=a^2, \end{array}\right. \end{aligned}$$

where $N\ge 3$, $\Omega \subseteq \mathbb {R}^N$ is an exterior domain, i.e., $\Omega$ is an unbounded domain with $\mathbb {R}^N{\setminus }\overline{\Omega }$ non-empty and bounded, $a>0$, $2<p<2+\frac{4}{N}$, and $\lambda \in \mathbb {R}$ is Lagrange multiplier, which appears due to the mass constraint $\Vert u\Vert _{L^2(\Omega )}=a$. We use Brouwer degree, barycentric functions and minimax method to prove that for any $a>0$, there is a positive solution $u\in H^1_0(\Omega )$ for some $\lambda <0$ if $\mathbb {R}^N{\setminus }\Omega$ is contained in a small ball. In addition, if we remove the restriction on $\Omega$ but impose that $a>0$ is small, then we also obtain a positive solution $u\in H^1_0(\Omega )$ for some $\lambda <0$. If $\Omega$ is the complement of unit ball in $\mathbb {R}^N$, then for any $a>0$, we get a positive radial solution $u\in H^1_0(\Omega )$ for some $\lambda <0$ by Ekeland variational principle. Moreover, we use genus theory to obtain infinitely many radial solutions $\{(u_n,\lambda _n)\}$ with $\lambda _n<0$, $I_p(u_n)<0$ for $n\ge 1$ and $I_p(u_n)\rightarrow 0^-$ as $n\rightarrow \infty$, where $I_p$ is the corresponding energy functional.