Abstract
In this article, generalizations of the well-known Pohožaev and Pucci-Serrin identities and (as consequences of them) some non-existence results for Dirichlet problems with p (x)-Laplacian are obtained.
The main ideas within this paper as well as the most important results obtained during the study are presented in Section 1.
In Section 2, there are recalled the main results concerning the generalized Sobolev spaces (also known as Sobolev spaces with variable exponent)which will be used throughout this paper.
In Sections 3 and 4, we will give some necessary technical results regarding the differential calculus in generalized Sobolev spaces and the differentiability properties of superposition operators between generalized Sobolev spaces (in the Marcus-Mizel direction).
In Sections 5 and 6, a Pohožaev-type identity and a non-existence result obtained by Pucci and Serrin for a Dirichlet problem with p-Laplacian are generalized to a Dirichlet problem with p (x)-Laplacian. The Pohožaev identity and the non-existence Pohožaev’s result for Dirichlet problems with Laplacian are also generalized to the case of p (x)-Laplacian. In order to achieve this generalization, the technical results from Sections 3 and 4 are involved.
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This research was supported by the Romanian Ministry of Education and Research through CEEX program (Contract No. CEx 06-2-11-12/25.07.2006).
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Dinca, G., Isaia, F. Generalized Pohožaev and Pucci-Serrin identities and non-existence results for p(x)-Laplacian type equations. Rend. Circ. Mat. Palermo 59, 1–46 (2010). https://doi.org/10.1007/s12215-010-0001-7
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DOI: https://doi.org/10.1007/s12215-010-0001-7
Keywords
- Chain rule
- p (x)-Laplacian
- Generalized Pohožaev identity
- Generalized Pucci-Serrin identity
- Non-existence theorem