Electronic Journal of Differential Equations,
Conference 08 (2002), pp. 103-120

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

Authors: Pavel Drabek (Univ. of West Bohemia, Plzevn, Czech Republic)
 
Abstract:
   In this paper we study  Dirichlet
   boundary-value problems, for the $p$-Laplacian, of the form
   $$\displaylines{
   - \Delta_p u -\lambda_1 |u|^{p-2} u = f\quad\mbox{ in }\Omega,\cr
   u = 0 \quad\mbox{ on  } \partial \Omega,
   }$$
   where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth 
   boundary  $\partial \Omega$, $N \geq 1, p>1$, $f \in C (\bar{\Omega})$ 
   and  $\lambda_1 > 0$ 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 $1<p<2$ and the case $p>2$. 
   We also give the characterization of the right hand sides $f$ for 
   which the Dirichlet problem above is solvable and has multiple solutions.

Published October 21, 2002.
Math Subject Classifications: 35J60, 35P30, 35B35, 49N10.
Key Words: $p$-Laplacian; variational methods; PS condition; Fredholm alternative; upper and lower solutions.