Partial Differential Equations
Nonhomogeneous boundary value problems in Orlicz–Sobolev spaces
Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 15-20.

We study the nonlinear Dirichlet problem div(log(1+|u|q)|u|p2u)=λ|u|p2u+|u|r2u in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary, while p, q and r are real numbers satisfying p,q>1, p+q<min{N,r}, r<(NpN+p)/(Np). The main result of this Note establishes that for any λ>0 this boundary value problem has infinitely many solutions in the Orlicz–Sobolev space W01LΦ(Ω), where Φ(t)=0tlog(1+|s|q)|s|p2sds.

On étudie le problème de Dirichlet non linéaire div(log(1+|u|q)|u|p2u)=λ|u|p2u+|u|r2u dans Ω, u=0 sur ∂Ω, où Ω est un domaine borné, régulier et p, q, r sont des nombres réels tels que p,q>1, p+q<min{N,r}, r<(NpN+p)/(Np). Le résultat principal de cette Note montre que pour tout λ>0 ce problème admet une infinité de solutions dans l'espace d'Orlicz–Sobolev W01LΦ(Ω), où Φ(t)=0tlog(1+|s|q)|s|p2sds.

Published online:
DOI: 10.1016/j.crma.2006.11.020
Mihăilescu, Mihai 1; Rădulescu, Vicenţiu 1

1 University of Craiova, Department of Mathematics, Street A.I. Cuza No. 13, 200585 Craiova, Romania
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     title = {Nonhomogeneous boundary value problems in {Orlicz{\textendash}Sobolev} spaces},
     journal = {Comptes Rendus. Math\'ematique},
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Mihăilescu, Mihai; Rădulescu, Vicenţiu. Nonhomogeneous boundary value problems in Orlicz–Sobolev spaces. Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 15-20. doi : 10.1016/j.crma.2006.11.020.

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