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

top
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 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

top

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

top
  1. Etude des valeurs propres et de la résonance pour l’opérateur p -Laplacien, Thése de doctorat, U.L.B., 1987–1988. (1987–1988) 
  2. On the Fredholm alternative for the p -Laplacian, Proc. Amer. Math. Soc. 125 (1997), 3555–3559. (1997) MR1416077
  3. The Fredholm alternative at the first eigenvalue for the one-dimensional p -Laplacian, J. Differ. Equations 151 (1999), 386–419. (1999) MR1669705
  4. C 1 + d local regularity of weak solutions of degenerate elliptic equations, Nonlin. Anal. 7 (1983), 827–850. (1983) MR0709038
  5. Geometry of the energy functional and the Fredholm alternative for the p -Laplacian in more dimensions, (to appear). (to appear) 
  6. Generic Fredholm alternative for the one dimensional p -Laplacian, Nonlin. Differ. Equations Appl. 8 (2001), 285–298. (2001) MR1841260
  7. Fredholm alternative for the p -Laplacian in higher dimensions, (to appear). (to appear) MR1864314
  8. Nonlinear Differential Equations, Chapman & Hall/CRC, Boca Raton, 1999. (1999) MR1695376
  9. Quasilinear Elliptic Equations with Degenerations and Singularities, De Gruyter Series in Nonlinear Anal. and Appl. 5, Walter de Gruyter, Berlin, New York, 1997. (1997) MR1460729
  10. Resonance problems for the p -Laplacian, J. Funct. Anal. 169 (1999), 189–200. (1999) MR1726752
  11. A counterexample to the Fredholm alternative for the p -Laplacian, Proc. Amer. Math. Soc. 127 (1999), 1079–1087. (1999) MR1646309
  12. An improved Poincaré inequality and the p -Laplacian at resonance for p > 2 , Preprint. MR1895113
  13. Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609–623. (1970) MR0267269
  14. Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal. 12 (1998), 1203–1219. (1998) 
  15. On the equation d i v ( | u | p - 2 u ) + λ | u | p - 2 u = 0 , Proc. Amer. Math. Soc. 109 (1990), 157–164. (1990) Zbl0714.35029MR1007505
  16. On the Fredholm alternative for the p -Laplacian in one dimension, Preprint. MR1881394
  17. On the Fredholm alternative for the p -Laplacian at the first eigenvalue, Preprint. MR1896161
  18. On the number and structure of solutions for a Fredholm alternative with p -Laplacian, Preprint. MR1935641
  19. Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equations 51 (1984), 126–150. (1984) MR0727034

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.