  In this work, we study the existence of $W_0^{1, p(\cdot)}$-solutions to the
following boundary value problem involving the $p(\cdot)$-Laplacian operator:
  \begin{equation*}
  \left\lbrace
  \begin{array}{l}
  -\Delta_{p(x)}u+|\nabla u|^{q(x)}=\lambda g(x)u^{\eta(x)}+f(x),
\quad\textnormal{ in } \Omega, \\\qquad \,\,\,\,\,\quad \quad\qquad\quad u\geq
0, \quad\textnormal{ in } \Omega
  \qquad \,\,\,\,\,\quad \quad\qquad\quad u= 0, \,\,\quad \text{on }
\partial\Omega.
  \end{array}
  \right. \end{equation*}under appropriate ranges on the variable exponents. We
give assumptions on $f$ and $g$ in terms of the growth exponents $q$ and $\eta$
under which the above problem has a non-negative solution for all $\lambda >
0$.
