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Standing waves for nonlinear Schrödinger equations with a general nonlinearity. (English) Zbl 1132.35078

The paper under review deals with positive solutions in \(H^1(\mathbb{R}^N)\) of the equation
\[ \varepsilon^2\Delta v- V(x) v+ f(v)= 0, \]
where \(\varepsilon> 0\) is sufficiently small, and \(f\) satisfies that \(f(e^{i\theta}v)= e^{i\theta}f(v)\). These solutions correspond precisely to the so-called standing waves \(\psi(x, t)= \exp(iEt/h)v(x)\) of the nonlinear Schrödinger equation
\[ ih{\partial\psi\over\partial t}+ {h^2\over 2}\Delta\psi= V(x)+ f(\psi)= 0,\qquad (t,x)\in \mathbb{R}\times \mathbb{R}^N. \]
The authors explore some essential conditions that guarantee the existence of localized ground states \(v\). In particular, they are concerned with single-peak solutions \(v\) which concentrate around local minimum points of \(V\) as \(\varepsilon\to 0\), since then the corresponding standing waves are possible candidates for orbitally stable solutions. Their main result is as follows.
Theorem. Let \(N\geq 3\), and assume that
(V1) \(V\in C(\mathbb{R}^N,\mathbb{R})\), and \(V_0:= \text{inf}_{\mathbb{R}^N}V> 0\);
(V2) there exists a bounded domain \(O\) such that \(m:= \text{inf}_OV> \min_{\partial O} V\);
(F1) \(\lim_{t\to 0^+}f(t)/t= 0\);
(F2) there exists \(p\in(1, (N+ 2)/(N- 2))\) such that \(\limsup_{t\to\infty}f(t)/t^p<\infty\);
(F3) there exists \(T> 0\) such that \(mT^2/2< \int^T_0f(t)\,dt\).
Then, for any sufficiently small \(\varepsilon> 0\), there exists a solution \(v_\varepsilon> 0\) in \(H^1(\mathbb{R}^N)\) possesses the properties (i) there exists a maximum point \(x_\varepsilon\) of \(v_\varepsilon\) such that the distance from it to the domain
\[ \{x\in O: V(x)= m\} \] tends to \(0\) as \(\varepsilon\to 0\);
(ii) if \(\varepsilon\to 0\) and up to a subsequence, the functions \(w_\varepsilon(x):= v_\varepsilon(\varepsilon(x- x_\varepsilon))\) converge uniformly to a least energy solution \(u> 0\) in \(H^1(\mathbb{R}^N)\) of the equation
\[ \Delta u- mu+ f(u)= 0; \]
(iii) \(v_\varepsilon(x)\leq C\exp(-c|x- x_\varepsilon|/\varepsilon)\) for some constants \(c\), \(C> 0\) independent of \(\varepsilon\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
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