On S-shaped and reversed S-shaped bifurcation curves for singular problems

Abstract

We analyze the positive solutions to the singular boundary value problem ( −(|u ′ | p−2 u ′ ) ′ = � g(u) u� ; (0, 1), u(0) = 0 = u(1), where p > 1,� ∈ (0, 1),� > 0 and g : [0, ∞) → R is a C 1 function. In particular, we discuss examples when g(0) > 0 and wheng(0) < 0 that lead to S-shaped and reversed S-shaped bifurcation curves, respectively.