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 $$ -\Delta_p u=g(u),\qquad 1<p<n, $$ under a global critical Sobolev growth assumption and the monotonicity condition that $s^{-(p^*-1)}g(s)$ 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 $$ \Big(\sup_{B_R}u\Big)\Big(\inf_{B_{2R}}u\Big)^{p-1} \le C R^{p-n}. $$ 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.