© Y. Fu and X. Zhang. 2009
Received: 19 November 2008
Accepted: 11 February 2009
Published: 18 February 2009
1. Introduction and Main Results
Since the space and were thoroughly studied by Kováčik and Rákosník , variable exponent Sobolev spaces have been used in the last decades to model various phenomena. In , Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids.
3. The Proof of Theorem 1.1
Now we get the result.
and it is easy to get the result.
By Lemmas 3.2–3.6 above, and [7, Proposition 2.1 and Remark 2.1], we know that the functional has a sequence of critical values as Now we complete the proof.
This work is supported by Science Research Foundation in Harbin Institute of Technology (HITC200702) and The Natural Science Foundation of Heilongjiang Province (A2007-04).
- Kováčik O, Rákosník J: On spaces and . Czechoslovak Mathematical Journal 1991,41(116)(4):592–618.MATHGoogle Scholar
- Růžička M: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics. Volume 1748. Springer, Berlin, Germany; 2000:xvi+176.MATHGoogle Scholar
- Chabrowski J, Fu Y: Existence of solutions for -Laplacian problems on a bounded domain. Journal of Mathematical Analysis and Applications 2005,306(2):604–618. 10.1016/j.jmaa.2004.10.028MathSciNetView ArticleMATHGoogle Scholar
- Chabrowski J, Fu Y: Corrigendum to: "Existence of solutions for -Laplacian problems on a bounded domain". Journal of Mathematical Analysis and Applications 2006,323(2):1483. 10.1016/j.jmaa.2005.11.036MathSciNetView ArticleGoogle Scholar
- Fan X, Han X: Existence and multiplicity of solutions for -Laplacian equations in . Nonlinear Analysis: Theory, Methods & Applications 2004,59(1–2):173–188.MathSciNetMATHGoogle Scholar
- Mihăilescu M, Rădulescu V: A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proceedings of the Royal Society of London. Series A 2006,462(2073):2625–2641. 10.1098/rspa.2005.1633MathSciNetView ArticleMATHGoogle Scholar
- De Figueiredo DG, Ding YH: Strongly indefinite functionals and multiple solutions of elliptic systems. Transactions of the American Mathematical Society 2003,355(7):2973–2989. 10.1090/S0002-9947-03-03257-4MathSciNetView ArticleMATHGoogle Scholar
- Felmer PL: Periodic solutions of "superquadratic" Hamiltonian systems. Journal of Differential Equations 1993,102(1):188–207. 10.1006/jdeq.1993.1027MathSciNetView ArticleMATHGoogle Scholar
- Edmunds DE, Lang J, Nekvinda A: On norms. Proceedings of the Royal Society of London. Series A 1999,455(1981):219–225. 10.1098/rspa.1999.0309MathSciNetView ArticleMATHGoogle Scholar
- Edmunds DE, Rákosník J: Sobolev embeddings with variable exponent. Studia Mathematica 2000,143(3):267–293.MathSciNetMATHGoogle Scholar
- Fan X, Zhao Y, Zhao D: Compact imbedding theorems with symmetry of Strauss-Lions type for the space . Journal of Mathematical Analysis and Applications 2001,255(1):333–348. 10.1006/jmaa.2000.7266MathSciNetView ArticleMATHGoogle Scholar
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