Perturbation of the p-Laplacian by vanishing nonlinearities (in one dimension)

We treat the quasi-linear spectral problem {-(vertical bar u'vertical bar(p-2)u')' = lambda vertical bar u vertical bar(p-2)u + h(x, u), 0 < x < a; u(0) = u(a) = 0, (P) at resonance with vanishing nonlinearity h(x, u) = g(u)+ f (x), g(u) -> 0 as u -> +/-infinity, and f is an element of L-infinity(0, a), f not equivalent to 0, satisfying certain orthogonality-related hypotheses, 1 < p < 3, p not equal 2. The parameter.. R takes an arbitrary resonant value. We first establish the boundedness of the set of all weak solutions in the Sobolev space W-0(1,p) 0 (0, a), which then enables us to obtain an existence result by the Leray-Schauder degree theory. The boundedness is obtained from a very precise asymptotic estimate valid for large solutions to (P) which can be applied thanks to a sufficiently fast rate of decay g(u) -> 0 as u -> +/-infinity. (C) 2012 Elsevier Ltd. All rights reserved.