Ambrosetti-Prodi problem with degenerate potential and Neumann boundary condition

We study the degenerate elliptic equation $-\mathop{\rm div}(|x|^\alpha\nabla u) =f(u)+t\phi(x)+h(x)$inaboundedopenset$\Omega$ with homogeneous Neumann boundary condition, where $\alpha \in (0,2)$ and $f$ has a linear growth. The main result establishes the existence of real numbers $t_{\ast }$ and $t^{\ast }$ such that the problem has at least two solutions if $t\le t_{\ast }$, there is at least one solution if $t_{\ast }<t\le t^{\ast }$, and no solution exists for all $t>t^{\ast }$. The proof combines a priori estimates with topological degree arguments.