We study the wave inequality with a Hardy potential ∂ ttu− Δ u+λ |x|2u≥ |u|pin (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− 222 $\begin{array}{} \displaystyle \left(\frac{N-2}{2}\right)^2 \end{array}$, 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 p c ( 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.

中文翻译：

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外域具有Hardy势的非均匀波不等式的临界行为

我们研究具有Hardy势∂ttu−Δu +λ| x |2u≥| u | pin（0，∞）×Ω，$$ \ begin {array} {} \ displaystyle \ partial_ {tt} u的波不等式-{\ it \ Delta} u + \ frac {\ lambda} {| x | ^ 2} u \ geq | u | ^ p \ quad \ mbox {in}（0，\ infty）\ times {\ it \ Omega} ，\ end {array} $$，其中Ω是单位球的外部，in N，N≥2，p> 1和λ≥− N− 222 $ \ begin {array} {} \ displaystyle \ left（\ frac {N-2} {2} \ right）^ 2 \ end {array} $，在不均匀边界条件下α∂u∂ν（t，x）+βu（t，x）≥w（x）on （0，∞）×∂Ω，$$ \ begin {array} {} \ displaystyle \ alpha \ frac {\ partial u} {\ partial \ nu}（t，x）+ \ beta u（t，x）\ geq w（x）\ quad \ mbox {on}（0，\ infty）\ times \ partial {\ it \ Omega}，\ end {array} $$其中α，β≥0和（α，β）≠（ 0，0）。即，我们表明存在一个临界指数pc（N，λ）∈（1，∞]，如果1 <p <pc（N，λ），上述问题对于∫∂Ωw（x）dσ> 0的任何w∈L 1（∂Ω）都不允许全局弱解，而如果p> pc（N，λ），则该问题不允许对某些w> 0.据我们所知，在先前的工作中未考虑对在外部区域具有哈迪势能的波浪不等式的临界行为进行研究。本文还提到了一些未解决的问题。