  We study a stationary scattering problem related to the nonlinear Helmholtz
equation $-\Delta u - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N
\ge 3$ and $k>0$. For a given incident free wave $\varphi \in
L^\infty(\mathbb{R}^N)$, we prove the existence of complex-valued solutions of
the form $u=\varphi+u_{\text{sc}}$, where $u_{\text{sc}}$ satisfies the
Sommerfeld outgoing radiation condition. Since neither a variational framework
nor maximum principles are available for this problem, we use topological fixed
point theory and global bifurcation theory to solve an associated integral
equation involving the Helmholtz resolvent operator. The key step of this
approach is the proof of suitable a priori bounds.
