Global minimality conjecture for positive minimal-growth solutions of p-Laplacian-type equations

Let $1<p<\infty$, let $Q_V$ be the functional under consideration on a domain $\Omega$, and let $\Omega_1\Subset\Omega$ be a smooth open set. Denote by $\mathcal{M}_{\Omega,\overline{\Omega_1}}$ the class of positive solutions having minimal growth outside $\overline{\Omega_1}$. A positive global solution $u$ of

$$Q'_V(u)=0$$

in $\Omega$ is assumed to satisfy $u\in\mathcal{M}_{\Omega,\overline{\Omega_1}}$. The global minimality conjecture. If $p\neq 2$, then $u$ is a global minimal solution. This conjecture proposes that, unlike the linear case, minimal growth outside one compact region forces global minimality for positive solutions of the nonlinear equation. Its resolution status is not specified in the source.