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

Abstract

Abstract We study the wave inequality with a Hardy potential ∂ t t u − Δ u + λ | x | 2 u ≥ | u | p in  ( 0 , ∞ ) × Ω , $$\begin{array}{} \displaystyle \partial_{tt}u-{\it\Delta} u+\frac{\lambda}{|x|^2}u\geq |u|^p\quad \mbox{in } (0,\infty)\times {\it\Omega}, \end{array}$$ where Ω is the exterior of the unit ball in ℝ N , N ≥ 2, p > 1, and λ ≥ − N − 2 2 2 $\begin{array}{} \displaystyle \left(\frac{N-2}{2}\right)^2 \end{array}$ , under the inhomogeneous boundary condition α ∂ u ∂ ν ( t , x ) + β u ( t , x ) ≥ w ( x ) on  ( 0 , ∞ ) × ∂ Ω , $$\begin{array}{} \displaystyle \alpha \frac{\partial u}{\partial \nu}(t,x)+\beta u(t,x)\geq w(x)\quad\mbox{on } (0,\infty)\times \partial{\it\Omega}, \end{array}$$ 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 < p c (N, λ), the above problem admits no global weak solution for any w ∈ L 1(∂ Ω) with ∫ ∂Ω w(x) dσ > 0, while if p > p c (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.