## Advances in Differential Equations

### On the number of sign-changing solutions of a semiclassical nonlinear Schrödinger equation

#### 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].

#### Article information

Source
Adv. Differential Equations, Volume 12, Number 7 (2007), 737-758.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867433

Mathematical Reviews number (MathSciNet)
MR2331522

Zentralblatt MATH identifier
1207.35040

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B25: Singular perturbations

#### Citation

D'Aprile, Teresa; Pistoia, Angela. On the number of sign-changing solutions of a semiclassical nonlinear Schrödinger equation. Adv. Differential Equations 12 (2007), no. 7, 737--758. https://projecteuclid.org/euclid.ade/1355867433