Decay estimates for solutions to non-autonomous critical p-Laplace problems
Laura Baldelli, Umberto Guarnotta
TL;DR
The paper addresses the decay at infinity of positive solutions to the non-autonomous critical p-Laplacian inequality $0\le -\Delta_p u \le f(x)+\Lambda u^{p^*-1}$ in $\mathbb{R}^N$, where $f$ decays and $p^* = \frac{Np}{N-p}$ is the critical Sobolev exponent. It develops a framework of a priori bounds, regularity theory, and scaling, using the doubling lemma to control blow-up and establish a preliminary decay, then refines this with scaling arguments and weak type estimates to obtain precise decay rates. The main result provides a two-sided decay for $u$ and $|\nabla u|$, namely $u(x) \lesssim (1+|x|^{\frac{N-p}{p-1}})^{-1}$ and $|\nabla u(x)| \lesssim (1+|x|^{\frac{N-1}{p-1}})^{-1}$, together with matching lower bounds and gradient lower bounds in exterior regions; these hold under broad non-autonomous data and reduce to known optimal profiles in the purely critical/autonomous setting (e.g., Talenti-type profiles). The work extends prior results on purely critical problems to differential inequalities with decaying sources, offering robust decay control and regularity properties relevant for non-autonomous critical phenomena.
Abstract
We prove optimal decay estimates for positive solutions to elliptic p-Laplacian problems in the entire Euclidean space, when a critical nonlinearity with a decaying source term is considered. Also gradient decay estimates are furnished. Our results extend previous theorems in the literature, in which a purely critical reaction is treated. The technique is based on a priori estimates, regularity results, and rescaling arguments, combined with the doubling lemma.