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 $L^{\infty }$-estimate for solutions $u\in D^{s,p}({\mathbb{R}}^N)$ of the fractional $p$-Laplacian equation % $$ u\in D^{s,p}(\R^N): (-Δ_p)^s u=f(x,u) \quad\mbox{in }\R^N, $$ % of the form % $$\Vert u{\Vert }_{\infty }\le C\Phi (\Vert u{\Vert }_{\beta })$$ % for some $\beta >p$, where $\Phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a data independent function with ${\lim }_{s\to 0^{+}}\Phi (s)=0$. 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.