Global $$L^\infty $$ and Decay Estimate for Fractional p-Laplacian Equations in $$D^{s,p}({\mathbb R}^N)$$

Abstract In this paper we present a new global $${L^\infty }$$ L ∞ -estimate for solutions $$u\in D^{s,p}({\mathbb R}^N)$$ u ∈ D s , p ( R N ) of the fractional p -Laplacian equation $$ u\in D^{s,p}({\mathbb R}^N): (-\Delta _p)^s u=f(x,u) \quad \text{ in } {\mathbb R}^N, $$ u ∈ D s , p ( R N ) : ( - Δ p ) s u = f ( x , u ) in R N , of the form $$ \Vert u\Vert _{\infty }\le C \Phi (\Vert u\Vert _{\beta }) $$ ‖ u ‖ ∞ ≤ C Φ ( ‖ u ‖ β ) for some $$\beta > p$$ β > p , where $$\Phi : {\mathbb R}^+\rightarrow {\mathbb R}^+$$ Φ : R + → R + is a data independent function with $$\lim _{s\rightarrow 0^+}\Phi (s)=0$$ lim s → 0 + Φ ( s ) = 0 . The obtained