Abstract
We study the wave inequality with a Hardy potential
partial derivative(tt)u -Delta u + lambda/vertical bar x vertical bar(2) u >= vertical bar u vertical bar(p) in (0, infinity) x Omega,
where Omega is the exterior of the unit ball in R-N, N >= 2, p > 1, and lambda >= -(N-2/2)(2), under the inhomogeneous boundary condition
alpha partial derivative u/partial derivative v(t, x) + beta u(t, x) >= w(x) on (0, infinity) x partial derivative Omega,
where alpha, beta >= 0 and (alpha, beta) not equal (0, 0). Namely, we show that there exists a critical exponent p(c) (N , lambda) is an element of (1, infinity] for which, if 1 < p < p(c) (N, lambda), the above problem admits no global weak solution for any w is an element of L-1 (partial derivative Omega) with f(partial derivative Omega) w(x) d sigma > 0, while if p > p(c) (N, lambda), 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.