On a class of $p(x)$-Laplacian equations without any growth and Ambrosetti-Rabinowitz conditions

The aim of this paper is to establish the existence of nontrivial solutions for $p(x)$-Laplacian equations without any growth and Ambrosetti-Rabinowitz conditions. Employing the cutoff function approach, we show that auxiliary problem has at least one nontrivial solution. Furthermore, we obtain nontrivial solutions for original problems using De Giorgi iteration. The results presented here extend some recent contributions obtained for problems driven by the $p(x)$-Laplacian or even to more general differential operators.