Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

By G. Evéquoz, Tobias Weth. 2014

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We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation −Δu−k2u=Q(x)|u|p−2u,u∈W2,p(RN) with N≥3, 2(N+1)(N−1)<p<2NN−2 and nonnegative Q∈L∞(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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3a356635393a02599880c9356086739c1acb0295