We study the nonlinear boundary value problem in Ω, on ∂Ω, where is a bounded domain with smooth boundary, λ is a positive real number, and the continuous functions and q satisfy and for any and any . By analyzing the growth of the functions and q we prove in this Note several existence results in Sobolev spaces with variable exponents.
On étudie le problème non linéaire dans Ω, sur ∂Ω, où est un domaine borné et régulier, λ est un nombre réel positif et et q sont des fonctions continues telles que et pour tout et chaque . En étudiant la croissance des fonctions et q on obtient dans cette Note plusieurs résultats d'existence dans des espaces de Sobolev aux exposants variables.
Accepted:
Published online:
@article{CRMATH_2007__345_10_561_0, author = {Mih\u{a}ilescu, Mihai and Pucci, Patrizia and R\u{a}dulescu, Vicen\c{t}iu}, title = {Nonhomogeneous boundary value problems in anisotropic {Sobolev} spaces}, journal = {Comptes Rendus. Math\'ematique}, pages = {561--566}, publisher = {Elsevier}, volume = {345}, number = {10}, year = {2007}, doi = {10.1016/j.crma.2007.10.012}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2007.10.012/} }
TY - JOUR AU - Mihăilescu, Mihai AU - Pucci, Patrizia AU - Rădulescu, Vicenţiu TI - Nonhomogeneous boundary value problems in anisotropic Sobolev spaces JO - Comptes Rendus. Mathématique PY - 2007 SP - 561 EP - 566 VL - 345 IS - 10 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2007.10.012/ DO - 10.1016/j.crma.2007.10.012 LA - en ID - CRMATH_2007__345_10_561_0 ER -
%0 Journal Article %A Mihăilescu, Mihai %A Pucci, Patrizia %A Rădulescu, Vicenţiu %T Nonhomogeneous boundary value problems in anisotropic Sobolev spaces %J Comptes Rendus. Mathématique %D 2007 %P 561-566 %V 345 %N 10 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2007.10.012/ %R 10.1016/j.crma.2007.10.012 %G en %F CRMATH_2007__345_10_561_0
Mihăilescu, Mihai; Pucci, Patrizia; Rădulescu, Vicenţiu. Nonhomogeneous boundary value problems in anisotropic Sobolev spaces. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 561-566. doi : 10.1016/j.crma.2007.10.012. http://archive.numdam.org/articles/10.1016/j.crma.2007.10.012/
[1] On norms, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 455 (1999), pp. 219-225
[2] Sobolev embedding with variable exponent, Studia Math., Volume 143 (2000), pp. 267-293
[3] Existence and nonexistence results for anisotropic quasilinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 21 (2004), pp. 715-734
[4] On spaces and , Czechoslovak Math. J., Volume 41 (1991), pp. 592-618
[5] A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 462 (2006), pp. 2625-2641
[6] On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., Volume 135 (2007), pp. 2929-2937
[7] M. Mihăilescu, P. Pucci, V. Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., in press, | DOI
[8] Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer, Berlin, 1983
[9] On imbedding, continuation and approximation theorems for differentiable functions of several variables, Russian Math. Surveys, Volume 16 (1961), pp. 55-104
[10] Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Heidelberg, 1996
[11] Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., Volume 18 (1969), pp. 3-24
[12] On embeddings into Orlicz spaces and some applications, J. Math. Mech., Volume 17 (1967), pp. 473-483
[13] On embedding theorems for spaces of functions with partial derivatives of various degree of summability, Vestnik Leningrad. Univ., Volume 16 (1961), pp. 23-37
Cited by Sources: