A fibering map approach to a semilinear elliptic boundary value problem

We prove the existence of at least two positive solutions for the semilinear elliptic boundary-value problem $$ -Delta u(x) = lambda a(x) u^q + b(x) u^p quadmbox{for } x in Omega; quad u(x) = 0 quad mbox{for } x in partial Omega $$ on a bounded region $Omega$ by using the Nehari manifold and the fibering maps associated with the Euler functional for the problem. We show how knowledge of the fibering maps for the problem leads to very easy existence proofs.