Existence of solutions to n-dimensional pendulum-like equations

We study the elliptic boundary-value problem $$displaylines{ Delta u + g(x,u) = p(x) quad hbox{in } Omega cr uig|_{partial Omega} = hbox{ m constant}, quad int_{partialOmega} frac {partial u}{partial u} = 0, }$$ where $g$ is $T$-periodic in $u$, and $Omega subset mathbb{R}^n$ is a bounded domain. We prove the existence of a solution under a condition on the average of the forcing term $p$. Also, we prove the existence of a compact interval $I_p subset mathbb{R}$ such that the problem is solvable for $ilde p(x) = p(x) + c$ if and only if $cin I_p$.