Let p, q be functions on \(\mathbb {R}^{N}\) satisfying \(1\ll q\ll p\ll N\) , we consider p(x)-Laplacian problems of the form \(\begin{aligned} \left\{ \begin{array}{l} -\Delta _{p(x)}u+V(x)\vert u\vert ^{p(x)-2}u=\lambda \vert u\vert ^{q(x)-2}u+g(x,u)\text {,}\\ u\in W^{1,p(x)}(\mathbb {R}^{N})\text {.} \end{array} \right. \end{aligned}\) To apply variational methods, we introduce a subspace X of \(W^{1,p(x)}(\mathbb {R}^N)\) as our working space. Compact embedding from X into \(L^{q(x)}(\mathbb {R}^N)\) is established, this enable us to get nontrivial solution of the problem; and two sequences of solutions going to \(\infty \) and 0 respectively, when \(g(x,\cdot )\) is odd.
