Fredholm Alternative for the p-Laplacian I
by
Pavel Drabek
Centre of Applied Mathematics, University of West Bohemia, Plzen, Czech Republic
Coauthors: Manuel Del Pino (University of Chile), Petr Girg (University of West Bohemia), Gabriela Holubova (University of West Bohemia), Raul Manasevich (University of Chile), Peter Takac (University of Rostock), Michael Ulm (University of Rostock)

We shall discuss the existence and multiplicity of the solutions to the boundary value problem

\labeleqone-\Deltap u-\lambda|u|p-2u=f in \Omega     u=0 on \partial\Omega .

(\theequation)

Here \Omega subset R^N is a bounded domain, p > 1 real number and \lambda in R is a spectral parameter. Let \lambda₁ > 0 be the principal eigenvalue of -\Delta_p subject to homogeneous Dirichlet boundary conditions. We concentrate on the behaviour of the solutions under the assumption that \lambda is near \lambda₁ (and possibly \lambda = \lambda₁). In particular, we show that \int_\Omega f\phi₁=0 is sufficient condition for solvability of (). The variational approach combined with the method of upper and lower solutions is applied. We address several open problems. This talk will be predominantly theoretical with emphasis of general aspects of the methods used.

Date received: March 28, 2002