The Nehari manifold for a class of indefinite weight semilinear elliptic equations

Using the Nehari manifold, we prove the existence of positive solutions of the problem -Delta u = lambda a(x)u + b(x)vertical bar u vertical bar(gamma-2)u for x is an element of Omega, together with the boundary condition alpha u+(1-alpha)(partial derivative u/partial derivative n) = 0. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form t --> J(lambda)(u), where J(lambda) is the Euler functional associated with the equation) and a condition on b(x), we discuss how the Nehari manifold changes as lambda changes, and show how existence results for positive solutions of the equation are linked to properties of the Nehari manifold.