Complex Solutions and Stationary Scattering for the Nonlinear Helmholtz Equation

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