A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations. (English) Zbl 1126.35057

Ackermann, Nils

J. Funct. Anal. 234, No. 2, 277-320 (2006).

The author considers the following two Schrödinger equations \[ -\Delta u+V(x)u=f(x,u),\;u\in\text{H}^1(\mathbb{R}^N), \] where \(f\) is a Carathéodory function \(f:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}\). \[ -\Delta u+V(x)u=(W\star{u^2})u,\;u\in\text{H}^1(\mathbb{R}^3), \] where \(\star\) denotes convolution, and \(W\) denotes a measurable function \(W:\mathbb{R}^3\to[0,\infty).\) In an abstract setting he proves a nonlinear superposition principle for zeros of equivariant vector fields that are asymptotically additive in a well-defined sense. This result is used to obtain multibump solutions for two basic types of periodic stationary Schrödinger equations with superlinear nonlinearity. The nonlinear term may be of convolution type. If the superquadratic term in the energy functional is convex, the results also apply in certain cases if \(0\) is in a gap of the spectrum of the Schrödinger operator.

Reviewer: Thomas Ernst (Uppsala)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B10	Periodic solutions to PDEs

Keywords:

stationary nonlinear Schrödinger equation; convolution nonlinearity; periodic potential

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