This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: $${-div((1+frac{|{ abla}u|^{p(x)}}{sqrt{1+|{ abla}u|^{2p(x)}}}})|{ abla}u|^{p(x)-2}{ abla}u)={lambda}f(x,u);a.e.;in;{Omega}\u=0,;on;{partial}{Omega}$$ where ${Omega}{subset}R^N$ is a bounded domain with smooth boundary ${partial}{Omega}$, ${lambda}$ > 0 is a parameter. The purpose of this paper is to obtain the existence results of nontrivial solutions for every parameter ${lambda}$. Firstly, by using the mountain pass theorem a nontrivial solution is constructed for almost every parameter ${lambda}$ > 0. Then we consider the continuation of the solutions. Our results are a generalization of that of Manuela Rodrigues.