An exact multiplicity result for a class of symmetric problems

Abstract

We consider positive solutions of a class of semilinear problems

with even and positive $a(x)$, depending on a positive parameter $\lambda$. 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 $u$.