Revista Matemática Iberoamericana

Bound state solutions for a class of nonlinear Schrödinger equations

Denis Bonheure and Jean Van Schaftingen

Full-text: Open access

Abstract

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schr�dinger equations of the form $$ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, $ where $V, K$ are positive continuous functions and $p > 1$ is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential $V$ is allowed to vanish at infinity and the competing function $K$ does not have to be bounded. In the \emph{semi-classical limit}, i.e. for $\varepsilon\sim 0$, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function $\mathcal{A} = V^\theta K^{-\frac{2}{p-1}}$, where $\theta=(p+1)/(p-1)-N/2$. A special attention is devoted to the qualitative properties of these solutions as $\varepsilon$ goes to zero.

Article information

Source
Rev. Mat. Iberoamericana Volume 24, Number 1 (2008), 297-351.

Dates
First available in Project Euclid: 16 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1216247103

Mathematical Reviews number (MathSciNet)
MR2435974

Zentralblatt MATH identifier
1156.35084

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B25: Singular perturbations 35B40: Asymptotic behavior of solutions 35J10: Schrödinger operator [See also 35Pxx]

Keywords
nonlinear Schrödinger equation semi-classical states concentration vanishing potentials unbounded competition functions

Citation

Bonheure , Denis; Van Schaftingen , Jean. Bound state solutions for a class of nonlinear Schrödinger equations. Rev. Mat. Iberoamericana 24 (2008), no. 1, 297--351.https://projecteuclid.org/euclid.rmi/1216247103


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