  We prove the existence and asymptotic expansion of a large class of solutions
to nonlinear Helmholtz equations of the form \begin{equation*} (\Delta -
\lambda^2) u = N[u], \end{equation*} where $\Delta = -\sum_j \partial^2_j$ is
the Laplacian on $\mathbb{R}^n$ with sign convention that it is positive as an
operator, $\lambda$ is a positive real number, and $N[u]$ is a nonlinear
operator that is a sum of monomials of degree $\geq p$ in $u$, $\overline{u}$
and their derivatives of order up to two, for some $p \geq 2$. Nonlinear
Helmholtz eigenfunctions with $N[u]= \pm |u|^{p-1} u$ were first considered by
Guti\'errez. Such equations are of interest in part because, for certain
nonlinearities $N[u]$, they furnish standing waves for nonlinear evolution
equations, that is, solutions that are time-harmonic.
  We show that, under the condition $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for
every $f \in H^{k+2}(\mathbb{S}^{n-1})$ of sufficiently small norm, there is a
nonlinear Helmholtz function taking the form \begin{equation*} u(r, \omega) =
r^{-(n-1)/2} \Big( e^{-i\lambda r} f(\omega) + e^{+i\lambda r} g(\omega) +
O(r^{-\epsilon}) \Big), \text{ as } r \to \infty, \quad \epsilon > 0,
\end{equation*} for some $g \in H^{k}(\mathbb{S}^{n-1})$. Moreover, we prove
the result in the general setting of asymptotically conic manifolds.
