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Ackermann, Nils; Weth, Tobias

Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting. (English) Zbl 1070.35083

Commun. Contemp. Math. 7, No. 3, 269-298 (2005).
Summary: We prove the existence of infinitely many geometrically distinct two bump solutions of periodic superlinear Schrödinger equations of the type \(-\Delta u+ V(x)u= f(x,u)\), where \(x\in\mathbb R^N\) and \(\lim_{|x|\to\infty}u(x)= 0\). The solutions we construct change sign and have exactly two nodal domains. The usual multibump constructions for these equations rely on strong non-degeneracy assumptions. We present a new approach that only requires a weak splitting condition. In the second part of the paper we exhibit classes of potentials \(V\) for which this splitting condition holds.

Cited in 25 Documents

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J60 Nonlinear elliptic equations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces

Keywords:

existence; two bump solutions; periodic superlinear Schrödinger equations; splitting condition; standing waves; traveling waves; potential wells; equilibrium state
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References:

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