Oscillation Criteria For Bounded Solutions For Some Nonlinear Diffusion Equations Via Picone-Type Formulae

Abstract

By means of Picone's type identities and inequalities some comparison results for problems related to the equation $$\nabla .\{ A(u) \nabla u \} + c(x)u + f(u) = g(x) \text{ in } \mathbb R^n$$ are established.Because of its principal part,this type of equation finds its applications in various physical phenomena like in nonlinear diffusion problems,flows through porous media, plasma physics,...etc. In this paper we show how versatile can the use of Picone-type formulae be for these type of quasilinear equations. Our main focus is to establish some oscillation criteria for classical non trivial and bounded solutions of some of these types of equations.We will display here some criteria conditions for some model equations.The ultimate aim is that by means of comparison methods and some Picone-type formulae,this could lead to getting the oscillation criteria of some more general equations.