## Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity.(English)Zbl 1142.35082

The authors consider the nonlinear Schrödinger problem $$-\varepsilon ^{2}\Delta u+V(x)u=K(x)u^{p}$$, $$u>0$$, posed in $$\mathbb{R}^{n}$$, where $$\varepsilon >0$$, $$1<p<(n+2)/(n-2)$$. Here $$V$$ and $$K$$ are smooth bounded and positive potentials. The authors assume that $$A_{0}/(1+| x| ^{\alpha })\leq V(x)\leq A_{1}$$ and $$0\leq K(x)\leq k/(1+| x| ^{\beta })$$ on $$\mathbb{R}^{n}$$ with some hypotheses on $$\alpha$$ and $$\beta$$. They are looking for generalized solutions of this problem which belong to $$W^{1,2}(\mathbb{R}^{n})$$. They introduce the function $$Q(x)=V^{\theta }(x)K^{2/(p-1)}(x)$$, with $$\theta =(p+1)/(p-1)-n/2$$.
In the main result, the authors prove that if $$x_{0}$$ is an isolated stable stationary point of $$Q$$, for $$\varepsilon$$ small enough, then, the Schrödinger problem has a finite energy solution which concentrates at $$x_{0}$$. For the proof, the authors quote the previous result [see A. Ambrosetti, V. Felli and A. Malchiodi, J. Eur. Math. Soc. 7, No. 1, 117–144 (2005; Zbl 1064.35175)], where the existence was proved for the problem, assuming other hypotheses. The authors introduce some change of variables and a truncated nonlinearity which lead to an energy functional $$I_{\varepsilon }$$ defined on a Hilbert space $$E$$. The authors then use functional analysis arguments and intricate computations.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35D05 Existence of generalized solutions of PDE (MSC2000) 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

### Keywords:

nonlinear Schrödinger equations; bound states

Zbl 1064.35175
Full Text:

### References:

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