If \(N\in\mathbb{N}\) and \(\Omega\subseteq\mathbb{R}^N\) is a bounded domain with smooth boundary, consider the problem
\[ \begin{cases} -\text{div}(| \nabla u| ^{p(x)-2}\nabla u) +| u| ^{p(x)-2}u=\lambda f(x,u),&\text{in }\Omega,\\ | \nabla u| ^{p(x)-2}\frac{\partial u}{\partial\nu}= \mu g(x,u),&\text{on }\partial\Omega, \end{cases}\tag{1} \]
where \(p\in C(\overline{\Omega})\), \(p>1\) in \(\overline{\Omega}\), \(\lambda,\mu\in\mathbb{R}\), and \(\lambda^2+\mu^2>0\). Moreover, throughout the paper \(f\) and \(g\) denote Caratheodory functions with subcritical growth in a suitable sense, with respect to the variable exponent \(p(x)\). The author proves existence results that are analogues of classical results for the case \(p\equiv2\), using direct and minimax methods in a variational setting. The analogues for the following settings are covered: Sublinear and superlinear nonlinearities, superlinear odd nonlinearities, and odd concave-convex nonlinearities. Also the existence of a positive solution is considered in some of these cases.