Geodesic
The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems
Mathematica Bohemica, Tome 120 (1995) no. 2, pp. 169-195

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem \align-\operatorname{div}(a(x,u)|\nablau|^{p-2}\nabla u) = \lambda b(x,u)|u|^{p-2}u \quad\text{ in } \Omega, u = 0 \hskip2cm\text{ on } \partial\Omega, \endalign where $\Omega$ is a bounded domain, $p>1$ is a real number and $a(x,u)$, $b(x,u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a(x,u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^\infty(\Omega)$. The main tool is the investigation of the associated homogeneous eigenvalue problem and an application of the Schauder fixed point theorem.
We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem \align-\operatorname{div}(a(x,u)|\nablau|^{p-2}\nabla u) = \lambda b(x,u)|u|^{p-2}u \quad\text{ in } \Omega, u = 0 \hskip2cm\text{ on } \partial\Omega, \endalign where $\Omega$ is a bounded domain, $p>1$ is a real number and $a(x,u)$, $b(x,u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a(x,u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^\infty(\Omega)$. The main tool is the investigation of the associated homogeneous eigenvalue problem and an application of the Schauder fixed point theorem.
DOI : 10.21136/MB.1995.126227
Classification : 35B35, 35B45, 35J20, 35J65, 35J70, 35P30, 47H12, 47N20
Keywords: boundedness of eigenfunction; weighted Sobolev space; Schauder fixed point theorem; degenerated quasilinear partial differential equations; weak solutions; eigenvalue problems; boundedness of the solution 
Drábek, Pavel. The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems. Mathematica Bohemica, Tome 120 (1995) no. 2, pp. 169-195. doi: 10.21136/MB.1995.126227
@article{10_21136_MB_1995_126227,
     author = {Dr\'abek, Pavel},
     title = {The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems},
     journal = {Mathematica Bohemica},
     pages = {169--195},
     year = {1995},
     volume = {120},
     number = {2},
     doi = {10.21136/MB.1995.126227},
     mrnumber = {1357600},
     zbl = {0839.35049},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.1995.126227/}
}
TY  - JOUR
AU  - Drábek, Pavel
TI  - The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems
JO  - Mathematica Bohemica
PY  - 1995
SP  - 169
EP  - 195
VL  - 120
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.1995.126227/
DO  - 10.21136/MB.1995.126227
LA  - en
ID  - 10_21136_MB_1995_126227
ER  - 
%0 Journal Article
%A Drábek, Pavel
%T The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems
%J Mathematica Bohemica
%D 1995
%P 169-195
%V 120
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.1995.126227/
%R 10.21136/MB.1995.126227
%G en
%F 10_21136_MB_1995_126227

[1] R. A. Adams: Sobolev Spaces. Academic Press, Inc., New York, 1975. | MR | Zbl

[2] A. Anane: Simplicité et isolation de la premiére valeur propre du p-laplacien avec poids. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 725-728. | MR | Zbl

[3] G. Barles: Remarks on uniqueness results of the first eigenvalue of the p-Laplacian. Ann. Fac. des Sc. de Toulouse IX (1988), no. 1, 65-75. | MR | Zbl

[4] T. Bhattacharya: Radial symmetry of the first eigenfunction for the p-Laplacian in the ball. Proc. Amer. Math Society 104 (1988), 169-174. | MR | Zbl

[5] L. Boccardo: Positive eigenfunctions for a class of quasi-linear operators. Bollettino U. M. I. (5) 18-B (1981), 951-959. | MR | Zbl

[6] P. Drábek M. Kučera: Generalized eigenvalue and bifurcations of second order boundary value problems with jumping nonlinearities. Bull. Austral. Math. Society 37(1988), 179-187. | MR

[7] P. Drábek A. Kufner F. Nicolosi: On the solvability of degenerated quasilinear elliptic equations of higher order. J. Differential Equations 109 (1994), 325-347. | DOI | MR

[8] S. Fučík A. Kufner: Nonlinear Differential Equations. Elsevier, The Netherlands, 1980. | MR

[9] J. García Azorero I. Peral Alonso: Existence and non-uniqueness for the p-Laplacian: Non-linear eigenvalues. Comm. Partial Differential Equations 12 (1987), 1389-1430. | MR

[10] A. Kufner O. John S. Fučík: Function Spaces. Academia, Prague, 1977. | MR

[11] A. Kufner A. M. Sändig: Some Applications of Weighted Sobolev Spaces. Teubner, Band 100, Leipzig, 1987. | MR

[12] P. Lindqvist: On the equation $div(| \nabla u|^{p-2} \nabla u) + \lambda |u|^{p-2} u = 0$. Proc. Amer. Math. Society 109 (1990), 157-164. | MR | Zbl

[13] M. K. V. Murthy G. Stampacchia: Boundary value problems for some degenerate elliptic operators. Annali di Matematica (4) 80 (1968), 1-122. | MR

[14] M. Otani T. Teshima: The first eigenvalue of some quasilinear elliptic equations. Proc. Japan Academy 64, ser. A (1988), 8-10. | MR

[15] M. M. Vainberg: Variational Methods for the Study of Nonlinear Operators. Holden-Day, Inc. San Francisco, 1964. | MR | Zbl

Cité par Sources :