Weak Comparison Principle for Weighted Fractional p -Laplacian Equation

The aim of this paper is to establish a weak comparison principle for a class fractional p -Laplacian equation with weight. The nonlinear term f x , s > 0 is a Carathéodory function which is possibly unbounded both at the origin and at infinity and such that f x , s s 1 − p decreases with respect to s for a.e. x ∈ Ω .

A Picone identity for variable exponent operators and applications

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.

Maximum Principles for Dynamic Equations on Time Scales and Their Applications

We consider the second dynamic operators of elliptic type on time scales. We establish basic generalized maximum principles and apply them to obtain weak comparison principle for second dynamic elliptic operators and to obtain the uniqueness of Dirichlet boundary value problems for dynamic elliptic equations.

A Weak Comparison Principle for Reaction-Diffusion Systems

We prove a weak comparison principle for a reaction-diffusion system without uniqueness of solutions. We apply the abstract results to the Lotka-Volterra system with diffusion, a generalized logistic equation, and to a model of fractional-order chemical autocatalysis with decay. Moreover, in the case of the Lotka-Volterra system a weak maximum principle is given, and a suitable estimate in the space of essentially bounded functionsL∞is proved for at least one solution of the problem.

Symmetry Results for Nonvariational Quasi-Linear Elliptic Systems

By virtue of a weak comparison principle in small domains we prove axial symmetry in convex and symmetric smooth bounded domains as well as radial symmetry in balls for regular solutions of a class of quasi-linear elliptic systems in non-variational form. Moreover, in the two dimensional case, we study the system when set in a half-space.