NLS ground states on the half-line with point interactions (Q6039675)

The authors study ground states of the focusing nonlinear Schrödinger equation with a point singularity on the half-line \(\mathbb{R}_+=(0,\infty)\), i.e., the minimization problem \begin{align*} \mathcal{F}(\mu) := \inf_{u\in H_\mu^1(\mathbb{R}_+)}F(u), \tag{1} \end{align*} where \(\mu\in\mathbb{R}_+:=(0,\infty)\) denotes the mass of a potential minimizer, \begin{align*} H_\mu^1(\mathbb{R}_+) := \left\{u\in H^1(\mathbb{R}_+):\,\int_0^\infty |u|^2=\mu\right\}, \end{align*} and \begin{align*} F(u) := \frac12\int_0^\infty \left(|u'(x)|^2-\frac{|u(x)|^p}{p}\right)\,dx + \frac\alpha2|u(0)|^2. \end{align*} Here, \(\alpha\in\mathbb{R}\) denotes the strength of the point singularity at the origin and \(p\) belongs either to the interval \((2,6)\) (\(L^2\)-subcritical regime) or equals \(6\) (\(L^2\)-critical regime). The corresponding Euler--Lagrange equation is \[ \begin{cases} -u'' - |u|^{p-2}u = -\omega u & \quad \text{on}\ \mathbb{R}_+=(0,\infty), \\ u'(0) = \alpha u(0). \end{cases} \tag{2} \] The main results of the paper under review (Theorems~1.1, 1.2, 1.3, 1.5, 1.6 and Proposition 1.4) concern the existence and uniqueness of ground and bound states of (1) and (2), depending on the values of \(\alpha\), \(\mu\), and \(p\in(2,6]\). Using the notation \(\phi_\omega\) for the soliton solution to the stationary NLS \(-u''-|u|^{p-2}u=-\omega u\) on \(\mathbb{R}\), the results can be summarized as follows. \begin{itemize} \item[1.] (Theorem 1.1) If \(2<p<6\) and \(\alpha<0\), then for every \(\mu>0\) there is a unique positive ground state of (1). \item[2.] (Theorem 1.2) If \(2<p\leq4\) and \(\alpha>0\), then ground states of (1) exist if and only if \(\mu>\|\phi_{\alpha^2}\|_{L^2(\mathbb{R})}^2\). Moreover, when they exist, then the positive ground state is unique and coincides with the only positive bound state of (2) at mass \(\mu\). \item[3.] (Theorem 1.3) If \(4<p<6\) and \(\alpha>0\), then there is \(\mu^*=\mu^*(\alpha)<\|\phi_{\alpha^2}\|_{L^2(\mathbb{R})}^2\) such that bound states of (2) at mass \(\mu\) exist if and only if \(\mu\geq\mu^*\). Two positive bound states of mass \(\mu\) exist if and only if \(\mu\in(\mu^*,\|\phi_{\alpha^2}\|_{L^2(\mathbb{R})}^2)\). Moreover, there is \(\tilde \mu=\tilde\mu(\alpha)\) satisfying \(\tilde\mu\in(\mu^*,\|\phi_{\alpha^2}\|_{L^2(\mathbb{R})}^2)\) such that ground states of (1) at mass \(\mu\) exist if and only if \(\mu\geq\tilde\mu\). When they exist, the positive ground states are unique. \item[4.] (Proposition 1.4) There is an explicit function \(\gamma_p\) such that if \(p\leq 4\), then ground states of (1) at mass \(\mu\) exist if and only if \(\alpha<\gamma_p \mu^{\frac{p-2}{6-p}}\). If \(p\in(4,6)\), then ground states of (1) at mass \(\mu\) exist if and only if \(\alpha\leq\tilde h(\mu)\) with \(\tilde h(\mu)>\gamma_p \mu^{\frac{p-2}{6-p}}\). \item[5.] (Theorem 1.5) If \(p=6\) and \(\alpha<0\), then \(\mathcal{F}(\mu)=-c<0\) and ground states of (1) at mass \(\mu\) exist and coincide with the only positive bound state if \(\mu\in(0,\sqrt3\pi/4)\). On the other hand, \(\mathcal{F}(\mu)=-\infty\) if \(\mu\geq\sqrt3\pi/4\). \item[6.] (Theorem 1.6) If \(p=6\) and \(\alpha<0\), then \(\mathcal{F}(\mu)=0\) if \(\mu\in(0,\sqrt3\pi/4)\) and \(\mathcal{F}(\mu)=-\infty\) if \(\mu\geq\sqrt3\pi/4\). In particular, ground states of (1) at mass \(\mu\) do not exist for any \(\mu>0\). \end{itemize} The authors remark the following in view of existing literature. \begin{itemize} \item Theorem 1.1 was proved in greater generality, namely for star graphs with at least three half-lines, in [\textit{R. Adami} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 6, 1289--1310 (2014; Zbl 1304.81087)]. \item The occurrence of a phase transition at \(p=4\) for \(\alpha>0\) was also observed in [\textit{R. Fukuizumi} and \textit{L. Jeanjean}, Discrete Contin. Dyn. Syst. 21, No. 1, 121--136 (2008; Zbl 1144.35465)] in the context orbital stability of even bound states for the same NLS but on \(\mathbb{R}\); by symmetry, these results extend to the present situation on \(\mathbb{R}_+\). \item The content of Theorem 1.5 is mostly contained in [\textit{C. Cacciapuoti}, Contemp. Math. 717, 155--172 (2018; Zbl 1428.35487)]. \end{itemize} \smallskip Section~2 recalls some results concerning (1) and (2) with \(\alpha=0\). \smallskip Section~3 states and proves properties of bound states of (2), such as \begin{itemize} \item the threshold for existence or non-existence of bound states \(\eta^{\omega,\alpha}=\phi_\omega(\cdot-a)\) (where \(a=2\tanh^{-1}(\alpha/\omega)/[(p-2)\sqrt\omega]\)), depending on the ratio \(\omega/\alpha^2\) (Proposition~3.1), \item the dependence of the mass \(M(\omega,\alpha):=\|\eta^{\omega,\alpha}\|_2^2\) of bound states on \(\omega\) (Propositions~3.2, 3.8), \item the number \(|\mathcal{A_\mu}|:=|\{\eta^{\omega,\alpha}:\,M(\omega,\alpha)=\mu\}|\) of bound states with specific mass \(\mu>0\) (Corollaries~3.3, 3.9) depending on \(\alpha,p,\mu\), and \item a condition to decide which of two bound states has least energy when \(p\in(4,6)\) (Proposition~3.6). \end{itemize} \smallskip Sections~4 and 5 contains the proofs of the main results.