Summary: The aim of this paper is to establish an Ambrosetti-Prodi type result for the problem \[ \begin{cases} -\Delta u = K (|x|) f(|x|, u, |\nabla u|) +s\varphi, \quad x\in\Omega, \\ \alpha u + \beta \frac{\partial u}{\partial n}|_{\partial\Omega} = 0, \\ \lim_{|x|\to\infty} u(x)=0, \end{cases} \]where \(s\in\mathbb{R}\) is a parameter, \(\Omega=\{ x\in\mathbb{R}^N:|x|<r_0\}\), \(N\geq 3\), \(K:[r_0, \infty) \to [0, +\infty)\) is continuous and satisfies \(\int^\infty_{r_0} r^{N-1}K(r)dr <\infty\). \(f : [r_0, \infty) \times\mathbb{R}\times [0, +\infty) \to\mathbb{R}\) is continuous. \(\varphi\in C(\overline{\Omega})\) with \(\varphi \gneqq 0 \) in \(\omega\).