A Pohozaev-type neck proof of a conditional Harnack inequality in the critical $p$-Laplacian setting

Abstract

We prove a conditional Schoen-type Harnack inequality for positive weak solutions of the critical $p$-Laplace equation $$-{\Delta }_{p}u=g(u),1<p<n,$$ under a global critical Sobolev growth assumption and the monotonicity condition that $s^{-(p^{\ast }-1)}g(s)$ is nonincreasing. The result is conditional on two inputs, the classification of bounded positive entire blow-up limits as Aubin--Talenti $p$-bubbles and a preliminary singular-rate upper control on the normalized necks. Under these two hypotheses, solutions in $B_{3R}$ satisfy $$\left(\underset{B_R}{\sup }u\right)(\underset{B_{2R}}{\inf }u{)}^{p-1}\le C{R}^{p-n}.$$ The main point is a Pohozaev-neck argument which upgrades the preliminary singular decay rate $|x{|}^{-(n-p)/p}$ to the sharp $p$-harmonic fundamental rate $|x{|}^{-(n-p)/(p-1)}$. The argument replaces the Kelvin-transform and moving-sphere methods available in the conformally invariant semilinear case $p=2$, but unavailable for the general $p$-Laplacian.

26 pages