On Fredholm alternative for certain quasilinear boundary value problems

Drábek, Pavel

Mathematica Bohemica, Tome 127 (2002) no. 2, pp. 197-202.

Résumé

We study the Dirichlet boundary value problem for the $p$-Laplacian of the form \[ -\Delta _p u~- \lambda _1 |u|^{p-2} u~= f \ \text{in} \Omega ,\quad u~= 0 \ \text{on} \partial \Omega , \] where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary $\partial \Omega $, $ N \ge 1$, $ 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 \[ E_p(u) = \frac{1}{p} \int _{\Omega } |\nabla u|^p - \frac{\lambda _1}{p} \int _{\Omega } |u|^p - \int _{\Omega } fu \] and show the difference between the case $1$ 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.

Détail

DOI : 10.21136/MB.2002.134157

Classification : 35B35, 35J20, 35J60, 35P30, 47J30, 49N10
Keywords: $p$-Laplacian; variational methods; PS condition; Fredholm alternative; upper and lower solutions