Fractional p&q-Laplacian problems with potentials vanishing at infinity

Abstract

In this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional $p\&q$-Laplacian problems $\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}$ where $s \in (0,1)$, $1\lt p\lt q \lt\frac{N}{s}$, $V: \mathbb{R}^{N}\to \mathbb{R}$ and $K: \mathbb{R}^{N}\to \mathbb{R}$ are continuous, positive functions, allowed for vanishing behavior at infinity, $f$ is a continuous function with quasicritical growth and the leading operator $(-\Delta)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator.