On Fredholm alternative for certain quasilinear boundary value problems

Pavel Drábek

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 2, page 197-202
  • ISSN: 0862-7959

Abstract

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We study the Dirichlet boundary value problem for the p -Laplacian of the form - Δ p u - λ 1 | u | p - 2 u = f in Ω , u = 0 on ∂ Ω , where Ω ⊂ ℝ N is a bounded domain with smooth boundary ∂ Ω , N ≥ 1 , p > 1 , f ∈ C ( Ω ¯ ) and λ 1 > 0 is the first eigenvalue of Δ p . We study the geometry of the energy functional E p ( u ) = 1 p ∫ Ω | ∇ u | p - λ 1 p ∫ Ω | u | p - ∫ Ω 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 above Dirichlet problem is solvable and has multiple solutions.

How to cite

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Drábek, Pavel. "On Fredholm alternative for certain quasilinear boundary value problems." Mathematica Bohemica 127.2 (2002): 197-202. <http://eudml.org/doc/249060>.

@article{Drábek2002,
abstract = {We study the Dirichlet boundary value problem for the $p$-Laplacian of the form \[ -\Delta \_p u~- \lambda \_1 |u|^\{p-2\} u~= f \ \text\{in\} \Omega ,\quad u~= 0 \ \text\{on\} \partial \Omega , \] 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 $1<p<2$ 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.},
author = {Drábek, Pavel},
journal = {Mathematica Bohemica},
keywords = {$p$-Laplacian; variational methods; PS condition; Fredholm alternative; upper and lower solutions; -Laplacian; variational methods; PS condition; Fredholm alternative; upper and lower solutions},
language = {eng},
number = {2},
pages = {197-202},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Fredholm alternative for certain quasilinear boundary value problems},
url = {http://eudml.org/doc/249060},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Drábek, Pavel
TI - On Fredholm alternative for certain quasilinear boundary value problems
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 2
SP - 197
EP - 202
AB - 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 $1<p<2$ 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.
LA - eng
KW - $p$-Laplacian; variational methods; PS condition; Fredholm alternative; upper and lower solutions; -Laplacian; variational methods; PS condition; Fredholm alternative; upper and lower solutions
UR - http://eudml.org/doc/249060
ER -

References

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