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April, 2008 Bound state solutions for a class of nonlinear Schrödinger equations
Denis Bonheure , Jean Van Schaftingen
Rev. Mat. Iberoamericana 24(1): 297-351 (April, 2008).


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.


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Denis Bonheure . Jean Van Schaftingen . "Bound state solutions for a class of nonlinear Schrödinger equations." Rev. Mat. Iberoamericana 24 (1) 297 - 351, April, 2008.


Published: April, 2008
First available in Project Euclid: 16 July 2008

zbMATH: 1156.35084
MathSciNet: MR2435974

Primary: 35J60
Secondary: 35B25 , 35B40 , 35J10

Keywords: Concentration , nonlinear Schrödinger equation , semi-classical states , unbounded competition functions , vanishing potentials

Rights: Copyright © 2008 Departamento de Matemáticas, Universidad Autónoma de Madrid


Vol.24 • No. 1 • April, 2008
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