Summary: The variable exponent spaces are essential in the study of certain nonhomogeneous materials. In the framework of these spaces, we are concerned with a nonlinear elliptic problem involving a \(p(\cdot )\)-Laplace-type operator on a bounded domain \(\Omega \subset \mathbb R^{N}(N\geq 2)\) of smooth boundary \(\partial \Omega\). We introduce the variable exponent Sobolev space of the functions that are constant on the boundary and we show that it is a separable and reflexive Banach space. This is the space where we search for weak solutions to our equation \[ - \operatorname{div}(a(x,\nabla u))+|u|^{p(x)-2}u=\lambda f(x,u), \] provided that \(\lambda \geq 0\) and \(a:\overline{\Omega} \times \mathbb R^N\to \mathbb R^N, f:\Omega \times \mathbb R\to \mathbb R\) are fulfilling appropriate conditions. We use different types of mountain pass theorems, a classical Weierstrass type theorem and several three critical points theorems to establish existence and multiplicity results under different hypotheses. We treat separately the case when \(f\) has a \(p(\cdot)-1\)-superlinear growth at infinity and the case when \(f\) has a \(p(\cdot)-1\)-sublinear growth at infinity.