Nonexistence of local solutions to semilinear partial differential inequalities (Q1888395)

The authors investigate nonexistence of nonnegative solutions in any neighborhood of the origin, for semilinear elliptic inequalities of the following type: \[ -\Delta_x u\geq \lambda|x|^{-\mu}\cdot(x,\nabla u)+|x|^{-\alpha} u^q, \qquad u\geq 0\quad \text{in }\Omega,\tag{1} \] where \(\Omega\subset\mathbb{R}^e\), \(d\geq 3\) is a bounded smooth domain which contains the origin, \(q>1\) and \(\lambda\), \(\mu\), and \(\alpha\) are real parameters. Instead of using the maximum principle for the extended solutions, the authors take advantage of the local behavior of solutions near the origin by a direct bootstrap argument, which relies on a proper choice of the test functions.