  We consider the existence of solutions of the following $p(x)$-Laplacian
Dirichlet problem without the Ambrosetti-Rabinowitz condition:
$-\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \text{ in }\Omega,$ and
$u=0,\text{ on }\partial \Omega.$ We give a new growth condition and we point
out its importance for checking the Cerami compactness condition. We prove the
existence of solutions of the above problem via the critical point theory, and
also provide some multiplicity properties. Our results extend previous work by
Q. Zhang and C. Zhao, Existence of strong solutions of a $p(x)$-Laplacian
Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comp. Math.
Appl. 69 (2015), 1-12, and we establish the existence of solutions under weaker
hypotheses on the nonlinear term.
