A Critical Nonlocal Double Phase Problem in the Heisenberg Group With a Modified Hardy Potential

Abstract

In this article, we study a critical double phase problem involving fractional $(p,q)$ -sub-Laplacian $(-{\mathrm{\Delta }}_{\mathbb{H}}{)}_p^s+(-{\mathrm{\Delta }}_{\mathbb{H}}{)}_q^s$ with fractional exponent $0<s<1$ in the Heisenberg group ${\mathbb{H}}^N$ with homogeneous dimension $Q=2N+2$ as follows : $$\begin{array}{l}(-{\mathrm{\Delta }}_{\mathbb{H}}{)}_p^{s}u+(-{\mathrm{\Delta }}_{\mathbb{H}}{)}_q^{s}u+|u{|}^{p-2}u+|u{|}^{q-2}u-\sigma \psi \frac{|u{|}^{q-2}u}{r^{sq}}=\lambda f(\xi ,u)+|u{|}^{q_s^{*}-2}u\end{array}$$ where $1<p<q<Q$ and $f$ is a Carathéodory function with some growth assumptions, $\sigma \in \mathbb{R}$ and $\lambda >0$ are real parameters and $q_s^{*}=\frac{qQ}{Q-qs}$ is the fractional Folland–Stein–Sobolev critical exponent for $qs<Q$ . Here first we explore a suitable potential function $\psi$ and prove a modified Hardy’s inequality involving $\psi$ . We then give an application of this inequality by asserting a solution to the aforementioned problem.