A bifurcation problem governed by the boundary condition II

In this work we consider the problem Delta u = a(x)u(p) in Omega, partial derivative u/partial derivative nu = lambda u on partial derivative Omega, where Omega is a smooth bounded domain, nu is the outward unit normal to partial derivative Omega, lambda is regarded as a parameter and 0 < p < 1. We consider both cases where a(x) > 0 in Omega or a(x) is allowed to vanish in a whole subdomain Omega(0) of Omega. Our main results include existence of non-negative non-trivial solutions in the range 0 < lambda < sigma(1), where sigma(1) is characterized by means of an eigenvalue problem, uniqueness and bifurcation from infinity of such solutions for small lambda, and the appearance of dead cores for large enough lambda.