Nonlinear eigenvalue problems in Sobolev spaces with variable exponent
We study the boundary value problem -div((|∇u|p1(x)-2+|∇u|p2(x)-2)∇u)=f(x,u) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in ℝN. We focus on the cases when f±(x, u)=±(-λ|u|m(x)-2u+|u|q(x)-2u), where m(x)≔max{p1(x),p2(x)}<q(x)<N⋅m(x)N-m(x) for any x∈Ω̅. In the first case we show th...
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| Format: | Article |
| Language: | English |
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Hindawi Limited
2006-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2006/515496 |
| Summary: | We study the boundary value problem -div((|∇u|p1(x)-2+|∇u|p2(x)-2)∇u)=f(x,u) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in ℝN. We focus on the cases when f±(x, u)=±(-λ|u|m(x)-2u+|u|q(x)-2u), where m(x)≔max{p1(x),p2(x)}<q(x)<N⋅m(x)N-m(x) for any x∈Ω̅. In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods. |
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| ISSN: | 0972-6802 |