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

Abstract:
In this paper we study Dirichlet boundary-value problems, for the p-Laplacian, of the form
$$\displaylines{ - \Delta_p u -\lambda_1 |u|^{p-2} u = f\quad\hbox{ in }\Omega,\cr u = 0 \quad\hbox{ on } \partial \Omega, }$$
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.

Show me the PDF file (258K), TEX file, and other files for this article.

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

Return to the table of contents for this conference.
Return to the EJDE web page