Multiple small solutions for p ( x ) -Schrödinger equations with local sublinear nonlinearities via genus theory

In this paper, we deal with the following $p(x)$-Schrödinger problem: \begin{equation*} \begin{cases} -\text{div}(|\nabla u|^{p(x)-2}\nabla u)+V(x)\left\vert u\right\vert ^{p(x)-2}u=f(x,u) & \hbox{in $\mathbb{R}^{N}$ ;} \\ u\in W^{1,p(x)}(\mathbb{R}^{N}), & \hbox{} \end{cases} \end{equation*} where the nonlinearity is sublinear. We present the existence of infinitely many solutions for the problem. The main tool used here is a variational method and Krasnoselskii's genus theory combined with the theory of variable exponent Sobolev spaces. We also establish a Bartsch–Wang type compact embedding theorem for the variable exponent spaces.