Abstract
Abstract In this work, we establish a new Picone identity for anisotropic quasilinear operators, such as the p(x)-Laplacian defined as div(|∇ u| p(x)−2 ∇ u). Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents. This new Picone identity can be also used to prove some accretivity property to a class of fast diffusion equations involving variable exponents. Using this, we prove for this class of parabolic equations a new weak comparison principle.
Highlights
Introduction and main resultsThe main aim of this paper is to prove a new version of the Picone identity involving quasilinear elliptic operators with variable exponent
Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents
The Picone identity is already known for homogeneous quasilinear elliptic as p-Laplacian with < p < ∞
Summary
The main aim of this paper is to prove a new version of the Picone identity involving quasilinear elliptic operators with variable exponent. The Picone identity is already known for homogeneous quasilinear elliptic as p-Laplacian with < p < ∞. Picone considers the homogeneous second order linear di erential system (a (x)u ) + a (x)u = (b (x)v ) + b (x)v =. Proved for di erentiable functions u, v ≠ the pointwise relation:. V v and in [2], extended (1.1) to the Laplace operator, i.e. for di erentiable functions u ≥ , v > one has. In [3], Allegretto and Huang extended (1.2) to the p-Laplacian operator with entiable functions v > and u ≥ we have p p−.
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