In this paper, we prove the existence and nonexistence results for positive solutions to semilinear elliptic boundary value problems, with concave nonlinearities inside a smooth bounded domain and on the boundary. Our approach relies on sub and supersolutions, as well as the Nehari manifold that may contain the critical points for the energy functional associated with the boundary value problem. The fibering method helps us to study the properties of the Nehari manifold.
"Positive solutions of semilinear elliptic eigenvalue problems with concave nonlinearities." Adv. Differential Equations 12 (12) 1415 - 1436, 2007.