The purpose of this paper is to investigate the existence of weak solutions for a class of nonlinear elliptic equation driven by the fractional p-Laplacian operator as follows: \(\begin{aligned} (-\Delta )^{\alpha }_{p} u + V(x) |u|^{p-2} u= f(x,u)\ \text {in}\ \mathbb {R}^{N}, \end{aligned}\) where \((-\Delta )^{\alpha }_{p}\) is the fractional p-Laplacian operator with \(0<\alpha<1<p<\infty \) , \(V\in C(\mathbb {R}^{N},\mathbb {R})\) may change sign and f is only locally defined near the origin with respect to u. Using variational methods, we obtain multiplicity results for the above-mentioned equations under some new, weak, and general assumptions on the potential V and the nonlinearty f(x, u). The results of this paper are new even in the fractional Laplacian case. Some examples are also given to illustrate our main theoretical results.
