Bounded perturbations of homogeneous quasilinear operators using bifurcations from infinity

This paper deals with existence results for the following nonlinear problem with the Dirichlet p-Laplacian Delta(p) in a bounded domain Q subset of R-N: -Delta(p)u = lambda\u\(p-2)u + h(x,u) in Omega, u = 0 on partial derivativeOmega. Here, Delta(p)u = def div(\delu\(p-2)delu), where pis an element ofe (1,infinity) is a fixed number, h equivalent to h(x,s) is a given function from Omega x R into R, and lambda is an element of R stands for a spectral parameter. We focus on lambda close to lambda(1), including the resonant case lambda = lambda(1). The nonlinearity h is assumed to be of Landesman-Lazer type, but we can deal with vanishing nonlinearities as well. Our asymptotic method substitutes the Lyapunov-Schmidt method in some sense. Unlike in the semilinear case P = 2, our method can treat more general nonlinearities if p not equal 2 (vanishing nonlinearities with very fast decay). (C) 2004 Elsevier Inc. All rights reserved.