Abstract
Let Omega be a bounded domain in R-N; N > 1 with a smooth boundary or Omega = (0,1). We study positive solutions to the boundary value problem of the form:
-Delta(p)u -Delta(q)u = lambda f(u) in Omega,
u = 0 on partial derivative Omega,
where q is an element of [2, p), lambda is a positive parameter, and f : [0, infinity) bar right arrow R is a class of C-1, non-decreasing and p-sublinear functions at infinity (i.e. lim(t ->infinity) f(t)/t(p-1) = 0) that are negative at the origin (semipositone). We discuss the existence of positive solutions for lambda >> 1. Further, when p = 4, q = 2, Omega = (0,1) and f (s) = (s + 1)(gamma) - 2; gamma is an element of (0,3), we provide the exact bifurcation diagram for positive solutions. In particular, we observe two positive solutions for a finite range of lambda and a unique positive solution for lambda >> 1.
