Abstract: In this paper, we study the exact multiplicity and bifurcation diagrams of positive solutions for a semipositone mean curvature problem with concave nonlinearity ??? ?  u0 √1?u0 2  0 = λf(u), x ∈ (?L, L), u(?L) = 0 = u(L), No.x 李晓东等: 带有凹非线性项的平均曲率半正问题正解的确切个数 11 where λ and L are positive parameters, f ∈ C2 ([0, ∞), R) satisfies f(0) < 0 and f is concave for 0 < u < L. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.

Key words: Minkowski space, Semipositone, Positive solution, Time map, Exact multiplicity