A system of wave inequalities with inverse-square potentials in an exterior domain
This article is concerned with the existence and nonexistence of solutions to the coupled wave inequalities \(u_{tt}-\Delta u+\frac{\lambda}{|x|^2}u\geq |x|^a |v|^p\) for \((t, x)\in (0, \infty)\times B_1^c\) and \(v_{tt}-\Delta v+\frac{\lambda}{|x|^2}v\geq |x|^b |u|^q\) for \((t, x)\in (0, \infty)\times B_1^c\). Here \(B_1^c\) denotes the complement of the open unit ball in \(\mathbb{R}^N\), \(N\geq 2\), \(p, q>1\) and \(\lambda\geq - (N-2)^2/4\). Also, it is assumed that \((u, v)\) satisfies \(u\geq f\) and \(v\geq g\) on \((0,\infty)\times \partial B_1\). The two main results of the article provide necessary and suffient conditions for existence and nonexistence of solutions. Some particular cases are also included in the analysis. The approach is technical and relies on various integral estimates. One particular point to note is the careful selection of test functions in order to derive the nonexistence of a solution.
scientific article; zbMATH DE number 7915841
scientific article