The Behavior of Solutions to the Dirichlet Problem for Second Order Elliptic Equations with Variable Nonlinearity Exponent in a Neighborhood of a Conical Boundary Point

Abstract

We study the Dirichlet problem for the p-Laplacian in a conical domain with the homogeneous boundary condition on the lateral surface of a cone with vertex at the origin. We assume that the variable exponent p = p(x) is separated from 1 and ∞ and denote by Ω the intersection of the cone with the unit (n − 1)-dimensional sphere. We prove that (i) if p satisfies the Lipschitz condition and ∂Ω is of class C 2+β, then the solution to the Dirichlet problem is O(|x| λ ) in a neighborhood of the origin, where λ is the sharp exponent of tending to zero of solutions to the same Dirichlet problem for the p(0)-Laplacian and (ii) if p satisfies the Holder condition, p(0) = 2, and ∂Ω is of class C 1+β, then the solution to the Dirichlet problem is O(|x| λ0) in a neighborhood of the origin, where λ 0 is the sharp exponent of tending to zero of solutions to the same Dirichlet problem for the Laplace operator. Bibliography: 18 titles.