An exact multiplicity result for a class of symmetric problems

Abstract

We consider positive solutions of a class of semilinear problems

u'' + λa(x)f(u) = 0,   −1 < x < 1, u(−1) = u(1) = 0,

with even and positive a(x), depending on a positive parameter λ. In case f(u) is convex, an exact multiplicity result was given in P. Korman, Y. Li and T. Ouyang [6]; see also P. Korman [4] for the details. It was observed by P. Korman and J. Shi [7] that convexity requirement can be relaxed for large u (see also [5]). We show that convexity requirement can also be relaxed for small u.