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,λ) \in H_0^1(Ω) \times \mathbb{R} \text{ such that } -Δu + λu = |u|^{p-2}u \text{ in } Ω\text{ and } \int_Ω|u|^2dx = μ, \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. $2+4/N<p<2^{\ast }$, 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 $\tau$.