Geometry of the energy functional and the Fredholm alternative for the p-Laplacian in higher dimensions
Pavel Drábek · DOAJ (DOAJ: Directory of Open Access Journals) · 2002
In this paper we study Dirichlet boundary-value problems, for the $p$-Laplacian, of the form $$displaylines{ - Delta_p u -lambda_1 |u|^{p-2} u = fquadmbox{ in }Omega,cr u = 0 quadmbox{ on } partial Omega, }$$ where $Omega subset mathbb{R}^N$ is a bounded domain with smooth boundary $partial Omega$, $N geq 1, p>1$, $f in C (Omega)$ and $lambda_1 > 0$ is the first eigenvalue of $Delta_p$. We study the geometry of the energy functional $$ E_p(u) = frac{1}{p} int_{Omega} |abla u|^p - frac{lambda_1}{p} int_{Omega} |u|^p - int_{Omega} f u $$ and show the difference between the case $1 2$. We also give the characterization of the right hand sides $f$ for which the Dirichlet problem above is solvable and has multiple solutions.