Infinitely Many Periodic Solutions for Variable Exponent Systems
© Xiaoli Guo et al. 2009
Received: 4 December 2008
Accepted: 14 April 2009
Published: 1 June 2009
The first eigenfunction is used to construct the subsolution of constant exponent problems successfully.
Because of the nonhomogeneity of -Laplacian, -Laplacian problems are more complicated than those of -Laplacian. Maybe the first eigenvalue and the first eigenfunction of -Laplacian do not exist (see ). Even if the first eigenfunction of -Laplacian exists, because of the nonhomogeneity of -Laplacian, the first eigenfunction cannot be used to construct the subsolution of -Laplacian problems.
There are many papers on the existence of periodic solutions for -Laplacian elliptic systems, for example [21–24]. The results on the periodic solutions for variable exponent systems are rare. Through a new method of constructing sub-supersolution, this paper gives the existence of infinitely many periodic solutions for problem .
2. Main Results and Proofs
At first, we give an existence of positive solutions for variable exponent systems on bounded domain via sub-super-solution method. The result itself has dependent value.
Note. In , the present author discussed the existence of solutions of , under the conditions that is radial, , and . Because of the nonhomogeneity of variable exponent problems, variable exponent problems are more complicated than constant exponent problems, and many results and methods for constant exponent problems are invalid for variable exponent problems. In many cases, the radial symmetric conditions are effective to deal with variable exponent problems. There are many results about the radial variable exponent problems (see [4, 14, 16]), but the following Theorem 2.1 does not assume any symmetric conditions.
We will establish.
which satisfy and , then possesses a positive solution (see ).
According to the comparison principle (see ), we can see that .
According to the comparison principle (see ), we can see that .
Partly supported by the National Science Foundation of China (10701066 & 10671084) and China Postdoctoral Science Foundation (20070421107) and the Natural Science Foundation of Henan Education Committee (2008-755-65 & 2009A120005) and the Natural Science Foundation of Jiangsu Education Committee (08KJD110007).
- Acerbi E, Mingione G: Regularity results for stationary electro-rheological fluids. Archive for Rational Mechanics and Analysis 2002,164(3):213–259. 10.1007/s00205-002-0208-7MathSciNetView ArticleMATHGoogle Scholar
- Acerbi E, Mingione G: Gradient estimates for the -Laplacean system. Journal für die Reine und Angewandte Mathematik 2005, 584: 117–148.MathSciNetView ArticleMATHGoogle Scholar
- Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics 2006,66(4):1383–1406. 10.1137/050624522MathSciNetView ArticleMATHGoogle 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
- Fan X: On the sub-supersolution method for -Laplacian equations. Journal of Mathematical Analysis and Applications 2007,330(1):665–682. 10.1016/j.jmaa.2006.07.093MathSciNetView ArticleMATHGoogle Scholar
- Fan X, Zhang Q, Zhao D: Eigenvalues of -Laplacian Dirichlet problem. Journal of Mathematical Analysis and Applications 2005,302(2):306–317. 10.1016/j.jmaa.2003.11.020MathSciNetView ArticleMATHGoogle Scholar
- El Hamidi A: Existence results to elliptic systems with nonstandard growth conditions. Journal of Mathematical Analysis and Applications 2004,300(1):30–42. 10.1016/j.jmaa.2004.05.041MathSciNetView ArticleMATHGoogle Scholar
- Harjulehto P, Hästö P, Latvala V: Harnack's inequality for -harmonic functions with unbounded exponent . Journal of Mathematical Analysis and Applications 2009,352(1):345–359. 10.1016/j.jmaa.2008.05.090MathSciNetView ArticleMATHGoogle Scholar
- Hudzik H: On generalized Orlicz-Sobolev space. Functiones et Approximatio Commentarii Mathematici 1976, 4: 37–51.MathSciNetMATHGoogle Scholar
- Mihăilescu M, Rădulescu V: Continuous spectrum for a class of nonhomogeneous differential operators. Manuscripta Mathematica 2008,125(2):157–167. 10.1007/s00229-007-0137-8MathSciNetView ArticleMATHGoogle Scholar
- Růžička M: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics. Volume 1748. Springer, Berlin, Germany; 2000:xvi+176.MATHGoogle Scholar
- Zhang Q: A strong maximum principle for differential equations with nonstandard -growth conditions. Journal of Mathematical Analysis and Applications 2005,312(1):24–32. 10.1016/j.jmaa.2005.03.013MathSciNetView ArticleMATHGoogle Scholar
- Zhang Q: Existence of solutions for -Laplacian equations with singular coefficients in . Journal of Mathematical Analysis and Applications 2008,348(1):38–50. 10.1016/j.jmaa.2008.06.026MathSciNetView ArticleMATHGoogle Scholar
- Zhang Q: Existence of positive solutions for elliptic systems with nonstandard -growth conditions via sub-supersolution method. Nonlinear Analysis: Theory, Methods & Applications 2007,67(4):1055–1067. 10.1016/j.na.2006.06.017MathSciNetView ArticleMATHGoogle Scholar
- Zhang Q: Existence and asymptotic behavior of positive solutions to -Laplacian equations with singular nonlinearities. Journal of Inequalities and Applications 2007, 2007:-9.Google Scholar
- Zhang Q: Boundary blow-up solutions to -Laplacian equations with exponential nonlinearities. Journal of Inequalities and Applications 2008, 2008:-8.Google Scholar
- Zhang Q, Liu X, Qiu Z: The method of subsuper solutions for weighted -Laplacian equation boundary value problems. Journal of Inequalities and Applications 2008, 2008:-19.Google Scholar
- Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Mathematics of the USSR-Izvestiya 1987, 29: 33–36. 10.1070/IM1987v029n01ABEH000958View ArticleMATHGoogle Scholar
- Hai DD, Shivaji R: An existence result on positive solutions for a class of -Laplacian systems. Nonlinear Analysis: Theory, Methods & Applications 2004,56(7):1007–1010. 10.1016/j.na.2003.10.024MathSciNetView ArticleMATHGoogle Scholar
- Chen C: On positive weak solutions for a class of quasilinear elliptic systems. Nonlinear Analysis: Theory, Methods & Applications 2005,62(4):751–756. 10.1016/j.na.2005.04.007MathSciNetView ArticleMATHGoogle Scholar
- Feng J-X, Han Z-Q: Periodic solutions to differential systems with unbounded or periodic nonlinearities. Journal of Mathematical Analysis and Applications 2006,323(2):1264–1278. 10.1016/j.jmaa.2005.11.039MathSciNetView ArticleMATHGoogle Scholar
- Manásevich R, Mawhin J: Periodic solutions for nonlinear systems with -Laplacian-like operators. Journal of Differential Equations 1998,145(2):367–393. 10.1006/jdeq.1998.3425MathSciNetView ArticleMATHGoogle Scholar
- Sun J, Ke Y, Jin C, Yin J: Existence of positive periodic solutions for the -Laplacian system. Applied Mathematics Letters 2007,20(6):696–701. 10.1016/j.aml.2006.07.010MathSciNetView ArticleMATHGoogle Scholar
- Tian Y, Ge W: Periodic solutions of non-autonomous second-order systems with a -Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2007,66(1):192–203. 10.1016/j.na.2005.11.020MathSciNetView ArticleMATHGoogle Scholar
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