Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials (Q1572868)

The author considers the Cauchy problem of the following nonlinear Schrödinger equations: \[ ih\psi_t=-\frac{h^2}{2m}\triangle \psi+V(x)\psi-\gamma|\psi|^{p-1}\psi,\quad t \geq 0,\;x \in \mathbb{R}^N, \tag{1} \] \[ \psi(0,x)=\psi_0(x),\quad x \in \mathbb{R}^n, \tag{2} \] where \(h\), \(m\), \(\gamma\) and \(p\) are positive constants. As his result and proof do not rely on the parameters \(h\), \(m\), \(\gamma\), he assumes that without loss of generality, \(h=m=\gamma=1.\) Let \[ H:=\left\{\varphi \in H^1(\mathbb{R}^N),\int_{\mathbb{R}^N}V(x)|\varphi|^2 dx < \infty \right\}. \] \(H\) becomes a Hilbert space, continuously embedded in \(H^1(\mathbb{R}^N)\), when endowed with the inner product \[ \langle \varphi,\varphi \rangle_H=\int_{\mathbb{R}^N}[\nabla \varphi \nabla \overline{\varphi}+(V-\inf V)\varphi\overline{\varphi}+ \varphi\overline{\varphi}]dx. \] Let \(\|\cdot\|_H\) be the associated norm and for any \(\mu \geq 0\) let \(S_\mu\) be the set of minimizers of the minimization problem \[ \inf_{\left\{u \in H,\int_{\mathbb{R}^N}|u|^2 dx=\mu \right\}} E(u), \] where \[ E(\varphi):=\int_{\mathbb{R}^N}\Biggl[\frac{1}{4}|\nabla \varphi|^2+\frac{1}{2}|V(x)\varphi|^2-\frac{1}{p+1}|\varphi|^{p+1}\Biggr] dx,\quad\varphi \in H. \] The author proves the following stability theorem by the method of Cazenave and Lions. Theorem. Assume that \(V\) satisfies \(\inf V >-\infty\), \(V(x) \rightarrow \infty\) as \(|x|\rightarrow \infty\) and for each \(|\alpha|\geq 2\), \(|D^{\alpha}V|\) is bounded. Let \(1<p<\frac{4}{N}\), \(\mu>0.\) Then for arbitrary \(\varepsilon\), there exists \(\sigma>0\) such that for any \(\psi_0 \in H,\) if \[ \inf_{u \in {\mathcal S}_{\mu}}\|\psi_0 -u \|_H < \sigma, \] then the solution \(\psi(t,\cdot)\) of the Cauchy problem (1),(2) satisfies \[ \inf_{u \in {\mathcal S}_{\mu}}\|\psi(t,\cdot)-u(\cdot) \|_H < \varepsilon,\quad \text{for all }t \geq 0. \] The author also mentions the physical background of this problem.