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
Abstract: We study the Dirichlet boundary value problem for the $p$-Laplacian of the form
-\Delta_p u - \lambda_1 |u|^{p-2} u = f in \Omega,\quad u = 0 on \partial\Omega,
where $\Omega\subset\R^N$ is a bounded domain with smooth boundary $\partial\Omega$, $ N \geq1$, $ p>1$, $ f \in C (\overline{\Omega})$ and $\lambda_1 > 0$ is the first eigenvalue of $\Delta_p$. We study the geometry of the energy functional
E_p(u) = \frac1p \int_{\Omega} |\nabla u|^p - \frac{\lambda_1}p \int_{\Omega} |u|^p - \int_{\Omega} fu
and show the difference between the case $1<p<2$ and the case $p>2$. We also give the characterization of the right hand sides $f$ for which the above Dirichlet problem is solvable and has multiple solutions.
Keywords: $p$-Laplacian, variational methods, PS condition, Fredholm alternative, upper and lower solutions
Classification (MSC 2000): 35J60, 35P30, 35B35, 49N10
Full text available as PDF (smallest), as compressed PostScript (.ps.gz) or as raw PostScript (.ps).
Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
