Elsevier

Advances in Mathematics

Volume 280, 6 August 2015, Pages 690-728
Advances in Mathematics

Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

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Abstract

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equationΔuk2u=Q(x)|u|p2u,uW2,p(RN) with N3, 2(N+1)(N1)<p<2NN2 and nonnegative QL(RN). We prove the existence of nontrivial solutions for periodic Q as well as in the case where Q(x)0 as |x|. In the periodic case, a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation. The solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation.

MSC

primary
35J20
secondary
35J05

Keywords

Nonlinear Helmholtz equation
Standing waves
Dual variational method
Nonvanishing

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