Advances in Differential Equations

On interacting bumps of semi-classical states of nonlinear Schrödinger equations

Xiaosong Kang and Juncheng Wei

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We study concentrated positive bound states of the following nonlinear Schr\"odinger equation: \[ h^2 \Delta u - V(x) u + u^p=0,\ \ \ u>0, \ \ x \in R^N , \] where $ p$ is subcritical. We prove that, at a local maximum point $x_0$ of the potential function $V(x)$ and for arbitrary positive integer $K (K>1)$, there always exist solutions with $K$ interacting bumps concentrating near $x_0$. We also prove that at a nondegenerate local minimum point of $V(x) $ such solutions do not exist.

Article information

Adv. Differential Equations, Volume 5, Number 7-9 (2000), 899-928.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35A05 35B50: Maximum principles 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]


Kang, Xiaosong; Wei, Juncheng. On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differential Equations 5 (2000), no. 7-9, 899--928.

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