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Under some assumptions on the function p(x), we obtain global gradient estimates for weak solutions of the p(x)-Laplacian type equation in ${ℝ}^N$.

In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations (−Δ)su+V(x)u=λf(x,u)inRN, $${(-\Delta )}^{s}u+V\left(x\right)u=\lambda f(x,u)\mathrm{in}{ℝ}^N,$$ where [...] (−Δ)su(x)=2limε→0∫RN∖Bε(X)u(x)−u(y)|x−y|N+2sdy,x∈RN ${(-\Delta )}^{s}u\left(x\right)=2\underset{\epsilon \to 0}{lim}{\int }_{{ℝ}^N\setminus B_{\epsilon }\left(X\right)}\frac{u\left(x\right)-u\left(y\right)}{|x-y{|}^{N+2s}}dy,x\in {ℝ}^N$ is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.