| Title:
|
On Fredholm alternative for certain quasilinear boundary value problems (English) |
| Author:
|
Drábek, Pavel |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
127 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
197-202 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
$p$-Laplacian |
| Keyword:
|
variational methods |
| Keyword:
|
PS condition |
| Keyword:
|
Fredholm alternative |
| Keyword:
|
upper and lower solutions |
| MSC:
|
35B35 |
| MSC:
|
35J20 |
| MSC:
|
35J60 |
| MSC:
|
35P30 |
| MSC:
|
47J30 |
| MSC:
|
49N10 |
| idZBL:
|
Zbl 1074.35035 |
| idMR:
|
MR1981524 |
| DOI:
|
10.21136/MB.2002.134157 |
| . |
| Date available:
|
2012-10-05T12:54:05Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134157 |
| . |
| Reference:
|
[1] A. Anane: Etude des valeurs propres et de la résonance pour l’opérateur $p$-Laplacien.Thése de doctorat, U.L.B., 1987–1988. |
| Reference:
|
[2] P. A. Binding, P. Drábek, Y. X. Huang: On the Fredholm alternative for the $p$-Laplacian.Proc. Amer. Math. Soc. 125 (1997), 3555–3559. MR 1416077, 10.1090/S0002-9939-97-03992-0 |
| Reference:
|
[3] M. Del Pino, P. Drábek, R. Manásevich: The Fredholm alternative at the first eigenvalue for the one-dimensional $p$-Laplacian.J. Differ. Equations 151 (1999), 386–419. MR 1669705, 10.1006/jdeq.1998.3506 |
| Reference:
|
[4] E. Di Benedetto: $C^{1+d}$ local regularity of weak solutions of degenerate elliptic equations.Nonlin. Anal. 7 (1983), 827–850. MR 0709038, 10.1016/0362-546X(83)90061-5 |
| Reference:
|
[5] P. Drábek: Geometry of the energy functional and the Fredholm alternative for the $p$-Laplacian in more dimensions.(to appear). |
| Reference:
|
[6] P. Drábek, P. Girg, R. Manásevich: Generic Fredholm alternative for the one dimensional $p$-Laplacian.Nonlin. Differ. Equations Appl. 8 (2001), 285–298. MR 1841260, 10.1007/PL00001449 |
| Reference:
|
[7] P. Drábek, G. Holubová: Fredholm alternative for the $p$-Laplacian in higher dimensions.(to appear). MR 1864314 |
| Reference:
|
[8] P. Drábek, P. Krejčí, P. Takáč: Nonlinear Differential Equations.Chapman & Hall/CRC, Boca Raton, 1999. MR 1695376 |
| Reference:
|
[9] P. Drábek, A. Kufner, F. Nicolosi: Quasilinear Elliptic Equations with Degenerations and Singularities.De Gruyter Series in Nonlinear Anal. and Appl. 5, Walter de Gruyter, Berlin, New York, 1997. MR 1460729 |
| Reference:
|
[10] P. Drábek, S. B. Robinson: Resonance problems for the $p$-Laplacian.J. Funct. Anal. 169 (1999), 189–200. MR 1726752, 10.1006/jfan.1999.3501 |
| Reference:
|
[11] P. Drábek, P. Takáč: A counterexample to the Fredholm alternative for the $p$-Laplacian.Proc. Amer. Math. Soc. 127 (1999), 1079–1087. MR 1646309, 10.1090/S0002-9939-99-05195-3 |
| Reference:
|
[12] J. Fleckinger-Pellé, P. Takáč: An improved Poincaré inequality and the $p$-Laplacian at resonance for $p>2$.Preprint. MR 1895113 |
| Reference:
|
[13] E. M. Landesman, A. C. Lazer: Nonlinear perturbations of linear elliptic boundary value problems at resonance.J. Math. Mech. 19 (1970), 609–623. MR 0267269 |
| Reference:
|
[14] G. Liebermann: Boundary regularity for solutions of degenerate elliptic equations.Nonlin. Anal. 12 (1998), 1203–1219. |
| Reference:
|
[15] P. Lindqvist: On the equation ${\mathrm div} (|\nabla u|^{p-2} \nabla u)+ \lambda |u|^{p-2} u=0$.Proc. Amer. Math. Soc. 109 (1990), 157–164. Zbl 0714.35029, MR 1007505, 10.1090/S0002-9939-1990-1007505-7 |
| Reference:
|
[16] R. Manásevich, P. Takáč: On the Fredholm alternative for the $p$-Laplacian in one dimension.Preprint. MR 1881394 |
| Reference:
|
[17] P. Takáč: On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue.Preprint. MR 1896161 |
| Reference:
|
[18] P. Takáč: On the number and structure of solutions for a Fredholm alternative with $p$-Laplacian.Preprint. MR 1935641 |
| Reference:
|
[19] P. Tolksdorf: Regularity for a more general class of quasilinear elliptic equations.J. Differ. Equations 51 (1984), 126–150. MR 0727034, 10.1016/0022-0396(84)90105-0 |
| . |
