On the number of positive solutions of singularly perturbed 1D NLS

• Autores: Patricio Luis Felmer Aichele, Salomé Martínez, Kazunaga Tanaka
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 8, Nº 2, 2006, págs. 253-268
• Idioma: inglés
• DOI: 10.4171/jems/51
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• Resumen

  • We study singularly perturbed 1D nonlinear Schrödinger equations (\ref{eq:1.1}). When $V(x)$ has multiple critical points, (\ref{eq:1.1}) has a wide variety of positive solutions for small $\varepsilon$ and the number of positive solutions increases to $\infty$ as $\varepsilon\to 0$. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V(x)$. Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.