Global $L^{\mathrm{\infty }}$ and decay estimate for fractional $p$-Laplacian equations in $D^{s,p}({\mathbb{R}}^N)$

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