Summary: In this paper, we investigate a Dirichlet problems with \(p(x)\)-Laplacian-like operators of the form \[ \begin{cases} -\operatorname{div}\left(|\nabla u|^{p(x)-2}\nabla u + \frac{|\nabla u|^{2p(x)-2}\nabla u}{\sqrt{1+|\nabla u|^{2p(x)}}}\right) = w|u|^{\varsigma(x) - 2}u + \varpi f(x, u, \nabla u) & \text{in }\Omega\\ u = 0 & \text{on }\partial\Omega \end{cases} \] in the setting of the variable-exponent Sobolev spaces \(W_0^{1, p(x)}(\Omega)\), where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) (\(N \geq 2\)), \(\omega\) and \(\varpi\) are two real parameters and \(p(\cdot), \varsigma(\cdot)\in C_+(\overline{\Omega})\). Under the suitable nonstandard growth conditions on \(f\) and using the topological degree for a class of demicontinuous operators of generalized (\(S_+\)) type, we establish the existence of “a weak solution” for the above problem.