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2007 On the number of sign-changing solutions of a semiclassical nonlinear Schrödinger equation
Teresa D'Aprile, Angela Pistoia
Adv. Differential Equations 12(7): 737-758 (2007).

Abstract

We study the following nonlinear Schrödinger equation: $$ {\varepsilon}^2\Delta u-V(x) u+|u|^{p-1}u=0 \ \hbox{in}\ \mathbb R^N,$$ where ${\varepsilon}>0$, $u,V:\mathbb R^N\to\mathbb R$, $p>1$. We prove that, under suitable conditions on the symmetry of $V$, the set of sign-changing solutions has a rich structure in the semiclassical limit ${\varepsilon}\to 0$: we construct multipeak solutions with an arbitrarily large number of positive and negative peaks which collapse to either a local minimum or a local maximum of $V.$ The proof relies on a local approach and is based on the finite-dimensional reduction, in the spirit of the arguments employed in [15].

Citation

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Teresa D'Aprile. Angela Pistoia. "On the number of sign-changing solutions of a semiclassical nonlinear Schrödinger equation." Adv. Differential Equations 12 (7) 737 - 758, 2007.

Information

Published: 2007
First available in Project Euclid: 18 December 2012

zbMATH: 1207.35040
MathSciNet: MR2331522

Subjects:
Primary: 35J60
Secondary: 35B25

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.12 • No. 7 • 2007
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