Harnack inequality for anisotropic fully nonlinear equations with nonstandard growth

Abstract

We establish Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear anisotropic elliptic equations with nonstandard growth, including anisotropic-$(p_i)$-Laplacian type equations with rough coefficients. The estimates hold under the explicit exponent condition: \begin{align*} p_n < \frac{\left(n- \fracλΛ\right)\bar{p}}{n-\fracλΛ\bar{p}}\quad \text{ or }\quad \bar{p} \geq\fracΛλn, \end{align*} where $\bar{p}$ is the harmonic mean of $2\le p_1\le \cdots \le p_n$. The key ingredient is a new max-type anisotropic barrier, constructed by maximizing over an auxiliary variable and balancing the different directional homogeneities. Combined with measure estimates and intrinsic-geometry arguments, this barrier yields the Harnack inequality. In the constant-coefficient case $\lambda =\Lambda$, we prove that the exponent condition is sharp for both weak and viscosity solutions. Using the same max-type approach, in the supercritical regime we construct a nontrivial nonnegative entire solution vanishing on a hyperplane, while at the critical threshold we construct positive entire solutions with arbitrarily large Harnack ratios. Finally, we exhibit a fully nonlinear counterexample showing that, for the general class, dependence of the admissible exponent range on the ellipticity ratio $\frac{\lambda }{\Lambda }$ cannot in general be removed.

arXiv comment30 pages. We improved the condition on (p_i) and determined the sharp exponent condition for the anisotropic-(p_i)-Laplacian