Picone's identity for a Finsler $p$-Laplacian and comparison of nonlinear elliptic equations

Abstract

In the paper we present an identity of the Picone type for a class of nonlinear differential operators of the second order involving an arbitrary norm $H$ in $\mathbb {R}^n$ which is continuously differentiable for $x \not = 0$ and such that $H^p$ is strictly convex for some $p > 1$. Two important special cases are the $p$-Laplacian and the so-called pseudo $p$-Laplacian. The identity is then used to establish a variety of comparison results concerning nonlinear degenerate elliptic equations which involve such operators. We also get criteria for the nonexistence of positive solutions in exterior domains for such equations by means of comparison with the equation exhibiting a kind of “anisotropic radial symmetry”.