Nonexistence of positive solutions and oscillation criteria for second- order semilinear elliptic inequalities (Q1894126)

The author studies the semilinear elliptic inequality \[ Lu + p(x) f(u) \leq 0 \text{ in } \Omega, \qquad \partial u/ \partial \nu \geq 0 \text{ on } \partial \Omega, \tag{*} \] where \(\Omega\) is an exterior domain and \(L\) an elliptic operator. He proves several nonexistence results for positive solutions for this problem. Furthermore several cases are studied, where solutions of (*) cannot be positive for large \(x \in \Omega\). The proofs of these results use comparison theorems, radial majorants and asymptotic estimates for ordinary differential equations.