The half-linear differential equation
is considered. Here , and is defined by , where is a fixed number. The authors continue the study initiated by A. Elbert [Colloq. Math. Soc. Janos Bolyai 30, 153-180 (1981; Zbl 0511.34006)]. They prove: Let , , and . If the boundary value problem
has a solution with on satisfying
then every solution of (E) must have a zero in unless and are proportional. Some oscillation and nonoscillation criteria are given, too, as applications.