In this paper, we study the following nonlinear elliptic problem involving the (p(y), q(y))-Laplacian operator: \(\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} -\text {div}(a(y)|\nabla v|^{p(y)-2} \nabla v) + b(y) |v|^{p(y)-2} v -\text {div}(|\nabla v|^{q(y)-2} \nabla v) & = g(y,v) \ \ & y \in \Omega , \\ & v = 0 \ \ & \text {on}\ \partial \Omega , \end{aligned} \end{array}\right. } \end{aligned}\) where \(\Omega \subset \mathbb {R}^n\) is a smooth bounded domain, \(1<q(y)<p(y)<n.\) We prove the existence of a weak solution in \(W^{1,p(y)}_{0}(\Omega )\) for the superlinear case and sublinear case by using the Mountain Pass Theorem.
