Modern Problems in Applied Analysis pp 1-14 | Cite as

# Boundary Value Problems for the Singular *p*- and *p*(*x*)-Laplacian Equations in a Cone

## Abstract

In this paper we describe briefly recent new results about the degenerate equations of the *p*-Laplacian type in a bounded cone. We shall consider the Dirichlet problem for such equation with the strong nonlinear right part as well as the Robin problem for such equation with singular nonlinearity in the right part. Such problems are mathematical models occurring in reaction-diffusion theory, non-Newtonian fluid theory, non-Newtonian filtration, the turbulent flow of a gas in porous medium, in electromagnetic problems, in heat transfer problems, in Fick’s law of diffusion et al. The aim of our investigations is the behavior of week solutions to the problem in the neighborhood of an angular or conical boundary point of the bounded cone. We establish sharp estimates of the type |*u*(*x*)| = *O*(|*x*|^{ϰ}) for the weak solutions *u* of the problems under consideration.

## Keywords

Degenerate equations*p*-Laplacian type bvp with strong nonlinear right part Angular or conical boundary point of the bounded cone Sharp estimate for week solution

## Mathematics Subject Classification (2010)

Primary 35J92 35G30; Secondary 35P20 76A05## References

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