Nonexistence of positive entire solutions for elliptic inequalities of the mean curvature type
The author gives conditions on \(p(x)\) and on \(f(u)\) that ensure the nonexistence of positive entire solutions of the equation \[ \text{div} \biggl[ {{Du} \over {(1+| Du|^ 2)^ \alpha}} \biggr]\geq p(x) f(u), \qquad x\in \mathbb{R}^ N \tag{\(*\)} \] with \(N\geq 2\), \(\alpha\in [0,1,2]\). The results extend those of the author's earlier paper [J. Math. Anal. Appl. 164, No. 1, 59-82 (1992; Zbl 0761.35025)]. The proofs are obtained by comparing with radial supersolutions. The reviewer remarks that although the author's statement that the capillarity equation is included in the general class \((*)\) is formally correct, that equation is excluded from the theorem by the conditions imposed on \(f(u)\). The result remains nevertheless correct for that case, as can be shown by an independent reasoning.
nonexistence of positive entire solutions
scientific article; zbMATH DE number 635894