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

TL;DR

This paper investigates the existence, multiplicity, and asymptotic behavior of solutions to a nonlinear Schrödinger equation with prescribed mass on bounded domains, covering subcritical, critical, and supercritical regimes.

Contribution

It provides exact counts of solutions, describes their limit behaviors, and extends results to specific domains and nonlinearities, including the case of the unit ball.

Findings

Infinitely many sign-changing solutions in subcritical case.

Precise asymptotic limits for solutions as mass tends to zero or infinity.

Exactly two positive solutions for small mass in supercritical regime on the unit ball.

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 $\tau$.