Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth

We investigate the following Dirichlet problem with variable exponents: \begin{equation*} \left\{ \begin{array}{l} -\bigtriangleup _{p(x)}u=\lambda \alpha (x)\left\vert u\right\vert ^{\alpha (x)-2}u\left\vert v\right\vert ^{\beta (x)}+F_{u}(x,u,v),\text{ in }\Omega , \\ -\bigtriangleup _{q(x)}v=\lambda \beta (x)\left\vert u\right\vert ^{\alpha (x)}\left\vert v\right\vert ^{\beta (x)-2}v+F_{v}(x,u,v),\text{ in }\Omega , \\ u=0=v,\text{ on }\partial \Omega. \end{array} \right. \end{equation*} We present here, in the system setting, a new set of growth conditions under which we manage to use a novel method to verify the Cerami compactness condition. By localization argument, decomposition technique and variational methods, we are able to show the existence of multiple solutions with constant sign for the problem without the well-known Ambrosetti--Rabinowitz type growth condition. More precisely, we manage to show that the problem admits four, six and infinitely many solutions respectively.