Existence of non-trivial solutions to a class of fractional p-Laplacian equations of Schrödinger-type with a combined nonlinearity

In this paper, we consider the following nonlinear fractional p-Laplacian equation of Schrödinger-type \(\begin{aligned} (-\Delta )_p^s u + V(x)|u|^{p-2}u =k(x){|u|}^ {\eta -2}u+ f(x,u), \ \ \qquad \ x \in \mathbb {R}^N \end{aligned}\) $$\begin{array}{c}\hfill {(-\mathrm{\Delta })}_p^{s}u+V(x){|u|}^{p-2}u=k(x){|u|}^{\eta -2}u+f(x,u),x\in {\mathbb{R}}^N\end{array}$$ where \(0<s<1<\eta <p\) $0<s<1<\eta <p$ , \(N\ge 2\) $N\ge 2$ , V(x) is a real continuous function on \(\mathbb {R}^N\) ${\mathbb{R}}^N$ . Based on some assumptions on k, V and f we obtain the existence of non-trivial solutions using of the variational methods.