Existence and asymptotics of nonlinear helmholtz eigenfunctions

Abstract

We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form (Δ-λ2)u = N[u], where Δ = Σj ∂ 2j is the Laplacian on Rn, λ is a positive real number, and N[u] is a nonlinear operator depending polynomially on u and its derivatives of order up to order two. Nonlinear Helmholtz eigenfunctions with N[u] = ±|u|p-1u were first considered by Gutierrez [Math. Ann., 328 (2004), pp. 1-25]. We show that for suitable nonlinearities and for every f ϵ Hk+4(Sn-1) of sufficiently small norm, there is a nonlinear Helmholtz function taking the form u(r,ω) = r-(n 1)/2(e -iλ r f(ω)+e+iλ rb(ω)+O(r -ϵ )), as r → ∞, ϵ > 0, for some b ∞ Hk(Sn-1). Moreover..