On Harnack's inequality for non-uniformly \(p\)-Laplacian equations (Q651838)

The paper deals with the elliptic equation in \(\Omega\) a domain in \({\mathbb R}^n\), \(n\geq 2\) \[ \text{div} (a(x) |\nabla u|^{p-2} \nabla u)=0, \] where \(p>1\), \(a\) is a positive measurable function on \(\Omega\) and there exist \(\lambda\) and \(\mu\) positive measurable function on \(\Omega\) such that \( \lambda (x) \leq a(x) \leq \mu (x) \) and \(\lambda \in L^t (\Omega)\) and \(\mu \in L^s(\Omega)\) with \( \frac{1}{t}+\frac{1}{s} < \frac{p}{n}\). The authors prove that if \(u\) is a non-negative weak solution of the equation and \(\Omega_0 \subset \subset \Omega\), then the Harnack inequality holds, i.e. \[ \sup_{\Omega_0} u \leq C \inf_{\Omega_0} u \] where \(C\) depends on \(p,n,t,s,\lambda, \mu,\Omega_0,\Omega\). The result is a generalization of a previos result given by \textit{N. S. Trudinger} [Arch. Ration. Mech. Anal. 42, 50--62 (1971; Zbl 0218.35035)].