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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

Citations:

Zbl 1072.35019
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References:

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