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, we prove the result in the general setting of asymptotically conic manifolds. The proof uses an elaboration of anisotropic Sobolev spaces defined by Vasy [A minicourse on microlocal analysis for wave propagation, in Asymptotic Analysis in General Relativity, London Math. Soc. Lecture Note Ser. 443, Cambridge University Press, Cambridge, 2018, pp. 219-374], between which the Helmholtz operator Δ λ2 acts invertibly. These spaces have a variable spatial weight I+-, varying in phase space and distinguishing between the two "radial sets"corresponding to incoming oscillations, e -iλr, and outgoing oscillations, e+iλr. Our spaces have, in addition, module regularity with respect to two different "test modules"and have algebra (or pointwise multiplication) properties that allow us to treat nonlinearities N[u] of the form specified above.