On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain

Abstract

We study the wave inequality with a Hardy potential (Eqation Presented), where Ω is the exterior of the unit ball in ℝN, N ≥ 2, p > 1, and λ ≥ - (N-2/2)2, under the inhomogeneous boundary condition 'Equation Presented', where α, β ≥ 0 and (α, β) ≠ (0, 0). Namely, we show that there exists a critical exponent pc(N, λ) ∈ (1, ∞] for which, if 1 < p < pc(N, λ), the above problem admits no global weak solution for any w ∈ L1 (∂Ω) with ∫∂Ω w(x) dσ > 0, while if p > pc(N, λ), the problem admits global solutions for some w > 0. To the best of our knowledge, the study of the critical behavior for wave inequalities with a Hardy potential in an exterior domain was not considered in previous works. Some open questions are also mentioned in this paper.