Abstract: By using the fountain theorem and the dual fountain theorem, respectively, the existence  and multiplicity of solutions for $p(x$)-Laplacian equations in $\mathbf{R}^{N}$ were studied, assumed that one of the perturbation terms $f_1(x,u),\, f_2(x,u)$ is superlinear and satisfies the Ambrosetti-Rabinowitz type condition and the other one is sublinear. The discussion was based on variable exponent Lebesgue and Sobolev spaces.

Key words: variable exponent Sobolev spaces, $p(x)$-Laplacian, (PS)$_c^\ast$ condition, fountain theorem, dual fountain theorem