This article addresses the following Dirichlet boundary value problem \(\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} -\Delta _{p(y)}v -\Delta _{q(y)}v +H(y)|v|^{s(y)-2}v & = g(y,v) \ \ & \text {in} \ \ \Omega , \\ v & = 0 \ \ & \text {on}\ \partial \Omega , \end{aligned} \end{array}\right. } \end{aligned}\) where \(\Omega \subset \mathbb {R}^N\) is a smooth bounded domain, \(1<q(y)<p(y)<N,\) H is an indefinite weight function that can change sign in \(\Omega \) and g(y, v) is a Carath \(\acute{e}\) odory function that satisfies some growth condition. Under appropriate conditions, the solution set may consist of a bounded infinite sequence of solutions or a unique solution by using the symmetric form of the Mountain Pass Theorem.
