## Concentration on curves for nonlinear Schrödinger equations.(English)Zbl 1123.35003

Suppose that $$p>1$$ and $$V\colon\mathbb R^N\to\mathbb R$$ is smooth and such that $$\inf_{x\in\mathbb R^N}V(x)>0$$. In the well known article [A. Ambrosetti, A. Malchiodi, and W.-M. Ni, Commun. Math. Phys. 235, No. 3, 427–466 (2003; Zbl 1072.35019)] it is proved that the problem $-\varepsilon^2 \Delta u + V(x) u=u^p\tag{1}$
possesses positive spike layer solutions concentrating near spheres of radius $$r_0$$ as $$\varepsilon\to0$$ if $$V$$ is radially symmetric and if the function $$r\mapsto r^{N-1}V^\sigma(r)$$ has a strict local extremum at $$r_0$$, with $$\sigma:=\frac{p+1}{p-1}-\frac12$$. Without imposing the condition of radial symmetry on $$V$$, the present work generalizes that result in the case $$N=2$$ (and proves a related conjecture exposed in that article) to the existence of positive solutions of (1) concentrating near nondegenerate closed geodesics of the weighted metric $$V^\sigma(dx_1^2+dx_2^2)$$ in $$\mathbb R^2$$. Here it has to be assumed that $$\varepsilon$$ is small enough and satisfies a gap condition
$| \varepsilon^2k^2-\lambda_*| \geq c\varepsilon,\qquad\forall k\in\mathbb N,$
with some constants $$c,\lambda_*>0$$. This condition implies bounds for the inverse of a linear differential operator that arises in the finite dimensional reduction.

### MSC:

 35B25 Singular perturbations in context of PDEs 35J60 Nonlinear elliptic equations 35Q55 NLS equations (nonlinear Schrödinger equations) 47F05 General theory of partial differential operators

### Keywords:

spike layer solution; semiclassical limit; standing wave

Zbl 1072.35019
Full Text:

### References:

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