Type: Article
Publication Date: 2020-01-01
Citations: 3
DOI: https://doi.org/10.1137/19m1307238
We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form $(\Delta - \lambda^2) u = N[u]$, where $\Delta = -\sum_j \partial^2_j$ is the Laplacian on $\mathbb{R}^n$, $\lambda$ 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 $N[u]= \pm |u|^{p-1} u$ were first considered by Gutiérrez [Math. Ann., 328 (2004), pp. 1--25]. We show that for suitable nonlinearities and for every $f \in H^{k+4}(\mathbb{S}^{n-1})$ of sufficiently small norm, there is a nonlinear Helmholtz function taking the form $u(r, \omega) = r^{-(n-1)/2} ( e^{-i\lambda r} f(\omega) + e^{+i\lambda r} b(\omega) + O(r^{-\epsilon})), \text{ as } r \to \infty, \quad \epsilon > 0$, for some $b \in H^{k}(\mathbb{S}^{n-1})$. 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 $\Delta - \lambda^2$ acts invertibly. These spaces have a variable spatial weight $\mathsf{l}_\pm$, varying in phase space and distinguishing between the two “radial sets” corresponding to incoming oscillations, $e^{-i\lambda r}$, and outgoing oscillations, $e^{+i\lambda r}$. 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.
