  We study the degenerate elliptic equation $-\mathop{\rm div}(|x|^\alpha\nabla
u) =f(u)+t\phi(x)+h(x)$ in a bounded open set $\Omega$ 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 $t_*$ and $t^*$ such that
the problem has at least two solutions if $t\leq t_*$, there is at least one
solution if $t_*<t\leq t^*$, and no solution exists for all $t>t^*$. The proof
combines a priori estimates with topological degree arguments.
