On Fredholm alternative for certain quasilinear boundary value problems
Mathematica Bohemica (2002)
- Volume: 127, Issue: 2, page 197-202
- ISSN: 0862-7959
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topDrábek, 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- Etude des valeurs propres et de la résonance pour l’opérateur -Laplacien, Thése de doctorat, U.L.B., 1987–1988. (1987–1988)
- 10.1090/S0002-9939-97-03992-0, Proc. Amer. Math. Soc. 125 (1997), 3555–3559. (1997) MR1416077DOI10.1090/S0002-9939-97-03992-0
- 10.1006/jdeq.1998.3506, J. Differ. Equations 151 (1999), 386–419. (1999) MR1669705DOI10.1006/jdeq.1998.3506
- 10.1016/0362-546X(83)90061-5, Nonlin. Anal. 7 (1983), 827–850. (1983) MR0709038DOI10.1016/0362-546X(83)90061-5
- Geometry of the energy functional and the Fredholm alternative for the -Laplacian in more dimensions, (to appear). (to appear)
- 10.1007/PL00001449, Nonlin. Differ. Equations Appl. 8 (2001), 285–298. (2001) MR1841260DOI10.1007/PL00001449
- Fredholm alternative for the -Laplacian in higher dimensions, (to appear). (to appear) MR1864314
- Nonlinear Differential Equations, Chapman & Hall/CRC, Boca Raton, 1999. (1999) MR1695376
- 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
- 10.1006/jfan.1999.3501, J. Funct. Anal. 169 (1999), 189–200. (1999) MR1726752DOI10.1006/jfan.1999.3501
- 10.1090/S0002-9939-99-05195-3, Proc. Amer. Math. Soc. 127 (1999), 1079–1087. (1999) MR1646309DOI10.1090/S0002-9939-99-05195-3
- An improved Poincaré inequality and the -Laplacian at resonance for , Preprint. MR1895113
- Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609–623. (1970) MR0267269
- Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal. 12 (1998), 1203–1219. (1998)
- 10.1090/S0002-9939-1990-1007505-7, Proc. Amer. Math. Soc. 109 (1990), 157–164. (1990) Zbl0714.35029MR1007505DOI10.1090/S0002-9939-1990-1007505-7
- On the Fredholm alternative for the -Laplacian in one dimension, Preprint. MR1881394
- On the Fredholm alternative for the -Laplacian at the first eigenvalue, Preprint. MR1896161
- On the number and structure of solutions for a Fredholm alternative with -Laplacian, Preprint. MR1935641
- 10.1016/0022-0396(84)90105-0, J. Differ. Equations 51 (1984), 126–150. (1984) MR0727034DOI10.1016/0022-0396(84)90105-0
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