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We study the behavior near the boundary conical point of weak solutions to the Dirichlet problem for elliptic quasi-linear second-order equation with the p -Laplacian and the strong nonlinearity on the right side.

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.

We study the behavior near the boundary conical point of weak solutions to the Dirichlet and Robin problems for elliptic quasi-linear second-order equation with the p-Laplacian and the strong nonlinearity in the right side, as well as we consider the Dirichlet problem for p(x)-harmonic functions. We establish the exact estimate of the solutions modulus for our problems near a conical boundary point of the type \(|u(x)|=O(|x|^{\varkappa })\).