MATHEMATICA BOHEMICA, Vol. 127, No. 2, pp. 197-202 (2002)

On Fredholm alternative for certain quasilinear boundary value problems

Pavel Drabek

Pavel Drabek, Centre of Applied Mathematics, University of West Bohemia, P. O. Box 314, 306 14 Plzen, Czech Republic, e-mail:

Abstract: We study the Dirichlet boundary value problem for the $p$-Laplacian of the form $$ -\Delta _p u - \lambda _1 |u|^{p-2} u = f \mbox { in } \Omega ,\quad u = 0 \mbox { on } \partial \Omega , $$ where $\Omega \subset \R ^N$ is a bounded domain with smooth boundary $\partial \Omega $, $ N \geq 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 $1<p<2$ 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 (MSC2000): 35J60, 35P30, 35B35, 49N10

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