Article
MSC:
35B35,
35J20,
35J60,
35P30,
47J30,
49N10 | MR 1981524 | Zbl 1074.35035 | DOI: 10.21136/MB.2002.134157
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
$p$-Laplacian; variational methods; PS condition; Fredholm alternative; upper and lower solutions
$p$-Laplacian; variational methods; PS condition; Fredholm alternative; upper and lower solutions
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.
References:
[1] A. Anane: Etude des valeurs propres et de la résonance pour l’opérateur $p$-Laplacien. Thése de doctorat, U.L.B., 1987–1988.
[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. DOI 10.1090/S0002-9939-97-03992-0 | MR 1416077
[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. DOI 10.1006/jdeq.1998.3506 | MR 1669705
[4] E. Di Benedetto: $C^{1+d}$ local regularity of weak solutions of degenerate elliptic equations. Nonlin. Anal. 7 (1983), 827–850. DOI 10.1016/0362-546X(83)90061-5 | MR 0709038
[5] P. Drábek: Geometry of the energy functional and the Fredholm alternative for the $p$-Laplacian in more dimensions. (to appear).
[6] P. Drábek, P. Girg, R. Manásevich: Generic Fredholm alternative for the one dimensional $p$-Laplacian. Nonlin. Differ. Equations Appl. 8 (2001), 285–298. DOI 10.1007/PL00001449 | MR 1841260
[7] P. Drábek, G. Holubová: Fredholm alternative for the $p$-Laplacian in higher dimensions. (to appear). MR 1864314
[8] P. Drábek, P. Krejčí, P. Takáč: Nonlinear Differential Equations. Chapman & Hall/CRC, Boca Raton, 1999. MR 1695376
[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
[10] P. Drábek, S. B. Robinson: Resonance problems for the $p$-Laplacian. J. Funct. Anal. 169 (1999), 189–200. DOI 10.1006/jfan.1999.3501 | MR 1726752
[11] P. Drábek, P. Takáč: A counterexample to the Fredholm alternative for the $p$-Laplacian. Proc. Amer. Math. Soc. 127 (1999), 1079–1087. DOI 10.1090/S0002-9939-99-05195-3 | MR 1646309
[12] J. Fleckinger-Pellé, P. Takáč: An improved Poincaré inequality and the $p$-Laplacian at resonance for $p>2$. Preprint. MR 1895113
[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
[14] G. Liebermann: Boundary regularity for solutions of degenerate elliptic equations. Nonlin. Anal. 12 (1998), 1203–1219.
[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. DOI 10.1090/S0002-9939-1990-1007505-7 | MR 1007505 | Zbl 0714.35029
[16] R. Manásevich, P. Takáč: On the Fredholm alternative for the $p$-Laplacian in one dimension. Preprint. MR 1881394
[17] P. Takáč: On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue. Preprint. MR 1896161
[18] P. Takáč: On the number and structure of solutions for a Fredholm alternative with $p$-Laplacian. Preprint. MR 1935641
[19] P. Tolksdorf: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equations 51 (1984), 126–150. DOI 10.1016/0022-0396(84)90105-0 | MR 0727034
Search
Partner of
