Existence and uniqueness of blow-up solutions for a class of logistic equations

Let f be a non-negative Cl-function on [0, infinity) such that f (u)/u is increasing and integral(1)(infinity) 1/rootF(t)dt < infinity, where F(t) = integral(0)(t) f(s)ds. Assume Omega subset of R-N is a smooth bounded domain, a is a real parameter and b greater than or equal to 0 is a continuous function on (Ω) over bar, b not equivalent to 0. We consider the problem Deltau + au = b(x)f (u) in Omega and we prove a necessary and sufficient condition for the existence of positive solutions that blow-up at the boundary. We also deduce several existence and uniqueness results for a related problem, subject to homogeneous Dirichlet, Neumann or Robin boundary condition.