From the introduction: For the following Neumann boundary value problem \[ -\Delta u=\mu_ku+ g(u)-h(x)\text{ in }\Omega, \]
\[ {\partial u\over \partial\eta} =0\text{ on }\partial \Omega, \] where \(\Omega\subset \mathbb{R}^N (N\geq 1)\) is a bounded domain with smooth boundary \(\partial\Omega\), \(\eta(x)\) denotes the outward normal, \(\mu_k\) is the \(k\)th distinct eigenvalue of the eigenvalue problem \[ -\Delta u=\mu u\text{ in }\Omega, \]
\[ {\partial u\over\partial \eta}= 0\text{ on }\partial \Omega, \] \(\mu_1=0\), \(g\in C(\mathbb{R},\mathbb{R})\), and \(h\in L^q (\Omega)\) for some \(q>2N/(N+2)\) if \(N\geq 3\) \((q>1\) if \(N=1,2)\), a new solvability condition is obtained by using the minimax methods in the critical point theory.