Are the maximum/minimum principles on $\mathbb{R}^n$ available for both local or nonlocal operators can be modified for a compact Riemannian manifold?
For instance, we have a nonlocal maximum principle for fractional $p$-Laplacian as follows (Brasco et al.) :
Let $1<p<\infty$ and $s\in(0,1)$ be such that $sp<N$. Let $\Omega\subset\mathbb{R}^N$ be a open bounded connected set, $a\in L^{N/sp}(\Omega)$ and let $u\in D_0^{s,p}(\Omega)\setminus\{0\}$ be a nonnegative function such that $$\begin{cases}(-\Delta)_p^su+au^{p-1}\geq\mu u^{q-1} \ \ \text{in} \ \ \Omega \\ u=0\hspace{4cm}\text{in} \ \ \mathbb{R}^N\setminus\Omega\end{cases}$$ for some $\mu\in\mathbb{R}$ and $p\leq q\leq\frac{Np}{N-ps}=:p_s^*$. Denote $\displaystyle D^{s,p}(\Omega):=\left\{u\in L^{p_s^*}(\Omega):\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dxdy<\infty\right\}$ and $D_0^{s,p}(\Omega):=\{u\in D^{s,p}(\Omega):u=0 \ \text{on} \ \mathbb{R}^N\setminus\Omega\}$. Then we have $u>0$ almost everywhere in $\Omega$.
I wonder if this result could be modified for some $u\in W^{s,p}(M)$ for $(M,g)$ a compact Riemannian $N$-manifold. So I thought by considering some cover of $(M,g)$ by some finite number of geodesic balls $B_\delta(x_i)$ for $x_i\in M$ and $0<\delta<\text{inj}(M)$ and taking the exponential chart, we can mimick the proof just as in the case of $\mathbb{R}^N$.
Will this approach work or there are some other ways to these kind of modifications? Any help is appreciated.