In this paper we present a new global \({L^\infty }\) -estimate for solutions \(u\in D^{s,p}({\mathbb R}^N)\) of the fractional p-Laplacian equation \( u\in D^{s,p}({\mathbb R}^N): (-\Delta _p)^s u=f(x,u) \quad \text{ in } {\mathbb R}^N, \) of the form \( \Vert u\Vert _{\infty }\le C \Phi (\Vert u\Vert _{\beta }) \) for some \(\beta > p\) , where \(\Phi : {\mathbb R}^+\rightarrow {\mathbb R}^+\) is a data independent function with \(\lim _{s\rightarrow 0^+}\Phi (s)=0\) . The obtained \(L^\infty \) -estimate is used to prove a decay estimate based on pointwise estimates in terms of nonlinear Wolff potentials. Taking advantage of both the \(L^\infty \) and decay estimate we prove a Brezis-Nirenberg type result regarding \(D^{s,2}({\mathbb R}^N)\) versus \(C_b\left( {\mathbb R}^N, 1+|x|^{N-2s}\right) \) local minimizers.
