2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems
Electron. J. Diff. Eqns., Conf. 08, 2002, pp. 103-120.

Geometry of the energy functional and the Fredholm alternative for the p-Laplacian in higher dimensions

Pavel Drabek

In this paper we study Dirichlet boundary-value problems, for the p-Laplacian, of the form
   - \Delta_p u -\lambda_1 |u|^{p-2} u = f\quad\hbox{ in }\Omega,\cr
   u = 0 \quad\hbox{ on  } \partial \Omega,
where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $N \geq 1$, 
p greater than 1, $f \in C (\bar{\Omega})$ and $\lambda_1$ greater than  0 is the first eigenvalue of $\Delta_p$. We study the geometry of the energy functional
   E_p(u) = \frac{1}{p} \int_{\Omega} |\nabla u|^p - \frac{\lambda_1}{p}
   \int_{\Omega} |u|^p - \int_{\Omega} f u
and show the difference between the case 1 less than p less than 2 and the case p greater than 2. We also give the characterization of the right hand sides $f$ for which the Dirichlet problem above is solvable and has multiple solutions.

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Pavel Drabek
Centre of Applied Mathematics
University of West Bohemia
P. O. Box 314, 306 14 Plzen, Czech Republic
e-mail: pdrabek@kma.zcu.cz

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