On $p(x)$-Laplacian equations in $\mathbb{R}^{N}$ with nonlinearity sublinear at zero

Abstract

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 \[ \left\{ \begin{array} [c]{l}% -\Delta_{p(x)}u+V(x)\vert u\vert ^{p(x)-2}u=\lambda\vert u\vert ^{q(x)-2}u+g(x,u)\text{,}\\ u\in W^{1,p(x)}(\mathbb{R}^{N})\text{.}% \end{array} \right. \] 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 proved, 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.