Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

الأصل · EN

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation -Δu-k² u =Q(x)|u|ᵖ⁻²u, u ∈ W²,ᵖ(Rⁿ) with N≥ 3, 2(N+1)/(N-1)< p<2N/N-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.