Exact number of positive solutions and existence of sign-changing solutions with prescribed mass for NLS on bounded domains

Abstract

Given $\mu > 0$, we study the elliptic problem: \begin{align*} \text{ find } (u,\lambda) \in H_0^1(\Omega) \times \mathbb{R} \text{ such that } -\Delta u + \lambda u = |u|^{p-2}u \text{ in } \Omega \text{ and } \int_\Omega|u|^2dx = \mu, \end{align*} where $\Omega \subset \mathbb{R}^N$ is a bounded domain and $p > 2$ is Sobolev-subcritical. When $p$ is $L^2$-subcritical, i.e. $2 < p < 2 + 4/N$, we show that the problem admits infinitely many sign-changing solutions whose energies are unbounded for every fixed $\mu > 0$. Moreover, we give the limit behavior for both the parameter $\lambda$ and the energy of the solutions as $\mu \to 0^+$ and $\mu \to +\infty$ respectively. Such a multiplicity result also holds when $p$ is $L^2$-critical, i.e. $p = 2 + 4/N$, for each small $\mu > 0$, and we describe precisely what happen when $\mu \to 0^+$. In the $L^2$-supercritical case, i.e. $2+4/N < p < 2^*$, we find as many sign-changing solutions as we want at the expense of possibly reducing the mass $\mu$. As $\mu$ tends to $0$, the energy of these solutions goes to $0$ and the limit of the parameter $\lambda$ is a Dirichlet eigenvalue of $-\Delta$ on $\Omega$ multiplying $-1$. When $\Omega = B_1$, the unitary ball, and the nonlinear term is $\tau |u|^{p-2}u$ with $\tau \in [1/2,1]$ fixed, in the $L^2$-supercritical regime, we prove that the problem admits exactly two positive solutions for small $\mu > 0$ and how small $\mu > 0$ must be does not depend on the value of $\tau$. Moreover, sending $\mu$ to $0$ we get that the energy of one positive solution tends to $0$ and the parameter tends to $-\lambda_1(B_1)$, where $\lambda_1(B_1)$ is the first Dirichlet eigenvalue of $-\Delta $ on the unit ball $B_1$, while both the energy of the other positive solution and the parameter $\lambda$ go to infinity uniformly with respect to $\tau$.