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

TL;DR

This paper establishes a new global $L^ Infty$-estimate for solutions of fractional $p$-Laplacian equations in $ R^N$, leading to decay estimates and a Brezis-Nirenberg type result comparing minimizers in different function spaces.

Contribution

Introduces a novel global $L^ Infty$-estimate for fractional $p$-Laplacian solutions and applies it to derive decay estimates and a Brezis-Nirenberg type theorem.

Findings

Established a new $L^ Infty$-estimate for solutions.

Derived decay estimates using nonlinear Wolff potentials.

Proved a Brezis-Nirenberg type result relating minimizers.

Abstract

In this paper we present a new global $L^\infty$-estimate for solutions $u\in D^{s,p}(\R^N)$ of the fractional $p$-Laplacian equation % $$ u\in D^{s,p}(\R^N): (-\Delta_p)^s u=f(x,u) \quad\mbox{in }\R^N, $$ % of the form % $$ \|u\|_{\infty}\le C \Phi(\|u\|_{\beta}) $$ % for some $\beta> p$, where $\Phi: \R^+\to \R^+$ 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 $D^{s,2}(\R^N)$ versus $C_b\left(\R^N, 1+|x|^{N-2s}\right)$ local minimizers.