Existence of solutions for critical systems with variable exponents
Abstract
In this work, we deal with elliptic systems under critical growth conditions on the nonlinearities. Using a variant of concentration-compactness principle, we prove an existence result.
Keyword : p(x)-Laplacian, generalized Sobolev spaces, critical Sobolev exponents, concentration-compactness principle, critical points theory
How to Cite
Lalilia, H., Tas, S., & Djellit, A. (2018). Existence of solutions for critical systems with variable exponents. Mathematical Modelling and Analysis, 23(4), 596-610. https://doi.org/10.3846/mma.2018.036
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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[16] X.L. Fan and Q.H. Zhang. Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear. Anal,52:1843–1852, 2003.
[17] X.L. Fan and D. Zhao. On the spacesLp(x)andWm,p(x).J. Math. Anal. Appl,263:424–446, 2001.
[18] Y. Fu. The principle of concentration compactness in L p(x)spaces and its application.Nonlinear. Anal,71(5-6):1876–1892, 2009.
[19] A. El Hamidi. Existence results to elliptic systems with nonstandard growth conditions. J. Math. Anal. App, 300:30–42, 2004.
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[26] S. Ogras, R.A. Mashiyev, M. Avci and Z. Yucedag. Existence of solution for aclass of elliptic systems in RN involving the (p(x),q(x))-Laplacian.J. Inequal. Appl., 612936:1–16, 2008.
[27] S. Samko. On a progress in the theory of Lebesgue spaces with variable exponent:maximal and singular operators. Integral Transforms Spec. Funct., 16(5):461–482, 2005.
[28] A. Silva. Multiple solutions for the p(x)-Laplace operator with critical growth. Adv. Nonlinear Stud., 11:63–75, 2011.
[29] M. Ruzicka. Electrorheological Fluids: Modeling and Mathematical theory. Springer-Verlage, Berlin, 2000.
[30] X. Xu and Y. An. Existence and multiplicity of solutions for elliptic systems with nonstandard growth conditions in RN. Nonlinear. Anal, 68:956–968, 2008.
[31] Q. Zhang, Y. Guo and G. Chen. Existence and multiple solutions for a variable exponent system. Nonlinear Anal., 73(12):3788–3804, 2010.
[32] X. Zhang and Y. Fu. Solutions of p(x)-Laplacian equations with critical exponent and perturbations in RN. EJDE, 2012(120):1–14, 2012.
[2] L. Boccardo and D.G. de Figueiredo. Some remarks on a system of quasilinear elliptic equations. Nonlinear Differen. Equat. and Appl, 9:309–323, 2002.
[3] J.F. Bonder, N. Saintier and A. Silva. Existence of solution to a critical equation with variable exponent. Ann. Acad. Sci. Fenn. Math., 37:579–594, 2012.
[4] J.F. Bonder, N. Saintier and A. Silva. On the Sobolev embedding theorem for variable exponent spaces in the critical range. JDE, 253(5):1604–1620, 2012.
[5] J.F. Bonder and A. Silva. Concentration-compactness principle for variable exponent spaces and applications. EJDE, 141:1–18, 2010.
[6] J.F. Bonder, S. Martınez and J.D. Rossi. Existence results for gradient elliptic systems with nonlinear boundary conditions.Nonlinear Differ. Equ. Appl., 14(1–2):153–179, 2007.
[7] H. Brezis and L. Nirenberg. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure. Appl. Math, 36:437–477, 1983.
[8] Y. Chen, S. Levine and M. Rao. Variable exponent, linear growth functionals in image restoration. SIAMJ. Appl. Math., 66(4):1383–1406, 2006.
[9] L. Diening, P. Harjulehto, P. Hasto and M. Ruzicka. Lebesgue and Sobolev spaces with variable exponents. Springer, 2010.
[10] L. Diening, P. Hasto and A. Nekvinda. Open problems in variable exponent Lebesgue and Sobolev spaces. FSDONA04 Proceedings, Milovy, Czech Republic, 66:38–58, 2004.
[11] A. Djellit and S. Tas. Quasilinear elliptic systems with critical Sobolev exponents in RN. Nonlinear Anal,66(7):1485–1497, 2007.
[12] A. Djellit, Z. Youbi and S. Tas. Existence of solutions for elliptic systems in RN involving the p(x)-aplacian. EJDE, 2012(131):1–10, 2012.
[13] D.E. Edmunds and J. Rakosnik. Sobolev embeddings with variable exponent. Studia Math.,143(3):267–293, 2000.
[14] X. L. Fan, Q. H. Zhang and D. Zhao. Eigenvalues of the p(x)-Laplacian Dirichlet problem.J. Math. Annl. App, 306(2):306–317, 2005.
[15] X.L. Fan, J. Shen and D. Zhao. Sobolev embedding theorems for spaces Wk,p(x)(Ω).J. Math. Anal. App., 262:749–760, 2001.
[16] X.L. Fan and Q.H. Zhang. Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear. Anal,52:1843–1852, 2003.
[17] X.L. Fan and D. Zhao. On the spacesLp(x)andWm,p(x).J. Math. Anal. Appl,263:424–446, 2001.
[18] Y. Fu. The principle of concentration compactness in L p(x)spaces and its application.Nonlinear. Anal,71(5-6):1876–1892, 2009.
[19] A. El Hamidi. Existence results to elliptic systems with nonstandard growth conditions. J. Math. Anal. App, 300:30–42, 2004.
[20] P. Hasto. The p(x)-Laplacian and applications.The Journal of Analysis, 15:53–62, 2007.
[21] H. Hudzik. On generalized Orlicz-Sobolev spaces. Funct. Approximatio Commentarii Mathematici, 4:37–51, 1976.
[22] Y. Jebri. The Mountain pass theorem. Combridge university press, New York, 2003.
[23] O. Kavian.Introduction a la theorie des points critiques et applications aux problemes elliptiques. Springer-Verlag France, Paris, 1993.
[24] F. Li, Z. Li and L. Pi. Variable exponent functionals in image restoration. App.Math. Comput., 216(3):870–882, 2010.
[25] P.L. Lions. The concentration-compactness principle in calculus of variation, the limit case part 1 and 2.Rev, Mat, Ibroamericana, 1(1):145–201, 1985.
[26] S. Ogras, R.A. Mashiyev, M. Avci and Z. Yucedag. Existence of solution for aclass of elliptic systems in RN involving the (p(x),q(x))-Laplacian.J. Inequal. Appl., 612936:1–16, 2008.
[27] S. Samko. On a progress in the theory of Lebesgue spaces with variable exponent:maximal and singular operators. Integral Transforms Spec. Funct., 16(5):461–482, 2005.
[28] A. Silva. Multiple solutions for the p(x)-Laplace operator with critical growth. Adv. Nonlinear Stud., 11:63–75, 2011.
[29] M. Ruzicka. Electrorheological Fluids: Modeling and Mathematical theory. Springer-Verlage, Berlin, 2000.
[30] X. Xu and Y. An. Existence and multiplicity of solutions for elliptic systems with nonstandard growth conditions in RN. Nonlinear. Anal, 68:956–968, 2008.
[31] Q. Zhang, Y. Guo and G. Chen. Existence and multiple solutions for a variable exponent system. Nonlinear Anal., 73(12):3788–3804, 2010.
[32] X. Zhang and Y. Fu. Solutions of p(x)-Laplacian equations with critical exponent and perturbations in RN. EJDE, 2012(120):1–14, 2012.