On interacting bumps of semi-classical states of nonlinear Schrödinger equations. (English) Zbl 1217.35065

Kang, Xiaosong; Wei, Juncheng

Adv. Differ. Equ. 5, No. 7-9, 899-928 (2000).

Summary: We study concentrated positive bound states of the following nonlinear Schrödinger equation \[ h^2 \Delta u \Gamma V (x)u + u^p = 0;~ u> 0;~ x\in\mathbb{R}^N \] 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.

MSC:

35J60	Nonlinear elliptic equations
35B50	Maximum principles in context of PDEs
35Q55	NLS equations (nonlinear Schrödinger equations)