Given a Brownian bridge B-0 with trend g : [0, 1] --> [0, infinity),
Y(z) = g(z) + B-0(z), z is an element of [0, 1], (1)
we are interested in testing H-0 : g equivalent to 0 against the alternative K : g > 0. For this test problem we study weighted Kolmogorov tests
reject H-0 double left right arrow sup(zis an element of[0,1]) w(z)Y(z) > c,
where c > 0 is a suitable constant and w : [0, 1] --> [0, infinity) is a weight function. To do such an investigation a recent result of the authors on a boundary crossing probability of the Brownian bridge is useful. In case the trend is large enough we show an optimality property for weighted Kohnogorov tests. Furthermore, an additional property for weighted Kolmogorov tests is shown which is useful to find the more favourable weight for specific test problems. Finally, we transfer our results to the change-point problem whether a regression function is or is not constant during a certain period.