On the Absence of Solutions for a Class of Singular Semilinear Differential Inequalities

The nonexistence of solutions to certain semilinear differential inequalities in a bounded domain is established. A model problem for such inequalities is $-\Delta u\ge |u|^q/|x|^\sigma$ ($\sigma \ge 2$) in a ball $B_R$. Similar evolution inequalities and fourth-order problems with the operator $\Delta ^2$ are analyzed. The proofs are based on the method of test functions.