Hyperbolic inequalities with a Hardy potential singular on the boundary of an annulus

Abstract

We are concerned with the study of existence and nonexistence of weak solutions for a class of hyperbolic inequalities with a Hardy potential singular on the boundary $ \partial B_1 $ of the annulus $ A = \left\{x\in \mathbb{R}^3: 1 < |x|\leq 2\right\} $, where $ \partial B_1 = \left\{x\in \mathbb{R}^3: |x| = 1\right\} $. A singular potential function of the form $ (|x|-1)^{-\rho} $, $ \rho\geq 0 $, is considered in front of the power nonlinearity. Two types of inhomogeneous boundary conditions on $ (0, \infty)\times \partial B_2 $, $ \partial B_2 = \left\{x\in \mathbb{R}^3: |x| = 2\right\} $, are studied: Dirichlet and Neumann. We use a unified approach to show the optimal criteria of Fujita-type for each case.

Keywords

• hyperbolic inequalities
• hardy potential
• singularities
• nonexistence