## 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)

### Keywords:

nonlinear Schrödinger equation; standing waves
Full Text:

### References:

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