Exact number of positive solutions and existence of sign-changing solutions with prescribed mass for NLS on bounded domains
Linjie Song, Wenming Zou
Abstract
Given $μ> 0$, 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 $Ω\subset \mathbb{R}^N$ 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 $μ> 0$. Moreover, we give the limit behavior for both the parameter $λ$ and the energy of the solutions as $μ\to 0^+$ and $μ\to +\infty$ respectively. Such a multiplicity result also holds when $p$ is $L^2$-critical, i.e. $p = 2 + 4/N$, for each small $μ> 0$, and we describe precisely what happen when $μ\to 0^+$. 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 $μ$. As $μ$ tends to $0$, the energy of these solutions goes to $0$ and the limit of the parameter $λ$ is a Dirichlet eigenvalue of $-Δ$ on $Ω$ multiplying $-1$. When $Ω= B_1$, the unitary ball, and the nonlinear term is $τ|u|^{p-2}u$ with $τ\in [1/2,1]$ fixed, in the $L^2$-supercritical regime, we prove that the problem admits exactly two positive solutions for small $μ> 0$ and how small $μ> 0$ must be does not depend on the value of $τ$. Moreover, sending $μ$ to $0$ we get that the energy of one positive solution tends to $0$ and the parameter tends to $-λ_1(B_1)$, where $λ_1(B_1)$ is the first Dirichlet eigenvalue of $-Δ$ on the unit ball $B_1$, while both the energy of the other positive solution and the parameter $λ$ go to infinity uniformly with respect to $τ$.