Abstract

We prove the existence and uniqueness of renormalized solutions to a class of nonlinear elliptic equations with variable exponents and measure data in L1(Ω)+W−1,p′(⋅)(Ω)L^{1}(\Omega)+W^{-1,p'(\cdot)}(\Omega).

Author information

Contact details are reproduced from the original publication and may be historical.

Boqiang Lv

Key Laboratory of Poyang Lake Environment and Resource Utilization, Nanchang University, Ministry of Education, Nanchang, 33047, China
and: College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, China

lbq86@yahoo.com.cn

Fengquan Li

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

fqli@dlut.edu.cn

Weilin Zou

College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, China

zwl267@yahoo.com.cn

Keywords

  • Nonlinear elliptic equations
  • variable exponents
  • measure data
  • renormalized solution
  • existence
  • uniqueness

Mathematics Subject Classification

35D05, 35J60, 35J70, 26D07

References

  1. E. Acerbi, G. Mingione: Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 156 (2001) 121–140.https://doi.org/10.1007/s002050100117Google Scholar
  2. E. Acerbi, G. Mingione: Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002) 213–259.https://doi.org/10.1007/s00205-002-0208-7Google Scholar
  3. S. N. Antontsev, S. I. Shmarev: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 60 (2005) 515–545.https://doi.org/10.1016/S0362-546X(04)00393-1Google Scholar
  4. M. Bendahmane, P. Wittbold: Renormalized solutions for nonlinear elliptic equations with variable exponents and L1 data, Nonlinear Anal. 70 (2009) 567–583.https://doi.org/10.1016/j.na.2007.12.027Google Scholar
  5. M. Bendahmane, P. Wittbold, A. Zimmermann: Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1 -data, J. Differ. Equations 249 (2010) 1483–1515.https://doi.org/10.1016/j.jde.2010.05.011Google Scholar
  6. P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vazquez: An L1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22 (1995) 240–273.Google Scholar
  7. A. Benkirane, J. Bennouna: Existence of solutions for nonlinear elliptic degenerate equations, Nonlinear Anal. 54 (2003) 9–37.https://doi.org/10.1016/S0362-546X(03)00031-2Google Scholar
  8. M. F. Betta, A. Mercaldo, F. Murat, M. M. Porzio: Existence of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side a measure, J. Math. Pures Appl. 82 (2003) 90–124.https://doi.org/10.1016/S0021-7824(03)00006-0Google Scholar
  9. D. Blanchard, F. Désir, O. Guibé: Quasi-linear degenerate elliptic problems with L1 data, Nonlinear Anal. 60 (2005) 557–587.https://doi.org/10.1016/j.na.2004.09.027Google Scholar
  10. D. Blanchard, F. Murat: Renormalized solutions of nonlinear parabolic problems with L1 data: existence and uniqueness, Proc. R. Soc. Edinb., Sect. A, Math. 127(6) (1997) 1137–1152.https://doi.org/10.1017/S0308210500026986Google Scholar
  11. D. Blanchard, F. Murat, H. Redwane: Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differ. Equations 177(2) (2001) 331–374.https://doi.org/10.1006/jdeq.2000.4013Google Scholar
  12. D. Blanchard, H. Redwane: Renormalized solutions for a class of nonlinear evolution problems, J. Math. Pures Appl. 77(2) (1998) 117–151.https://doi.org/10.1016/S0021-7824(98)80067-6Google Scholar
  13. L. Boccardo, D. Giachetti, J. I. Diaz, F. Murat: Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms, J. Differ. Equations 106 (1993) 215–237.https://doi.org/10.1006/jdeq.1993.1106Google Scholar
  14. B. K. Bonzi, S. Ouaro: Entropy solutions for a doubly nonlinear elliptic problem with variable exponent, J. Math. Anal. Appl. 370(2) (2010) 392–405.https://doi.org/10.1016/j.jmaa.2010.05.022Google Scholar
  15. B. Cekic, R. A. Mashiyev: Nontrivial solution for a nonlocal elliptic transmission problem in variable exponent sobolev spaces, Abstr. Appl. Anal. 2010 (2010), Art. ID 385048, 12 pp.https://doi.org/10.1155/2010/385048Google Scholar
  16. Y. Chen, S. Levine, M. Rao: Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006) 1383–1406.https://doi.org/10.1137/050624522Google Scholar
  17. G. Dal Maso, F. Murat, L. Orsina, A. Prignet: Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28(4) (1999) 741–808.Google Scholar
  18. R. J. Diperna, P. L. Lions: On the Cauchy problem for the Boltzmann equations: global existence and weak stability, Ann. Math. 130 (1989) 321–366.https://doi.org/10.2307/1971423Google Scholar
  19. X. Fan, J. Shen, D. Zhao: Sobolev embedding theorems for spaces W k,p(x) (Ω), J. Math. Anal. Appl. 262 (2001) 749–760.https://doi.org/10.1006/jmaa.2001.7618Google Scholar
  20. X. Fan, D. Zhao: On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001) 424–446.https://doi.org/10.1006/jmaa.2000.7617Google Scholar
  21. Y. Fu: Weak solution for obstacle problem with variable growth, Nonlinear Anal. 59 (2004) 371–383.https://doi.org/10.1016/S0362-546X(04)00264-0Google Scholar
  22. P. Harjulehto: Variable exponent Sobolev spaces with zero boundary values, Math. Bohem. 132 (2007) 125–136.https://doi.org/10.21136/MB.2007.134186Google Scholar
  23. P. Harjulehto, P. Hästö, M. Koskenoja, S. Varonen: The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal. 25(3) (2006) 205–222.https://doi.org/10.1007/s11118-006-9023-3Google Scholar
  24. D. Kinderlehrer, G. Stampacchia: An Introduction to Variational Inequalities and Their Applications, Academic Press, New York (1980).Google Scholar
  25. O. Kováčik, J. Rákosník: On spaces Lp(x) and W 1,p(x), Czech. Math. J. 41(116) (1991) 592–618.https://doi.org/10.21136/CMJ.1991.102493Google Scholar
  26. F. Li: Solvability of nonlinear elliptic boundary value problem with equivalued surface, Nonlinear Anal. 54 (2003) 995–1010.https://doi.org/10.1016/S0362-546X(03)00123-8Google Scholar
  27. J. L. Lions: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris (1969).Google Scholar
  28. R. A. Mashiyev, B. Cekic, O. M. Buhrii: Existence of solutions for p(x)-Laplacian equations, Electron. J. Qual. Theory Differ. Equ. 65 (2010), 13pp.https://doi.org/10.14232/ejqtde.2010.1.65Google Scholar
  29. S. Ouaro, S. Traoré: Weak and entropy solutions to nonlinear elliptic problems with variable exponent, J. Convex Analysis 16(2) (2009) 523–541.https://doi.org/10.68381/jca16029Google Scholar
  30. H. Redwane: Existence of a solution for a class of nonlinear parabolic systems, Electron. J. Qual. Theory Differ. Equ. 24 (2007), 18pp.https://doi.org/10.14232/ejqtde.2007.1.24Google Scholar
  31. M. Růžička: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics 1748, Springer, Berlin (2000).https://doi.org/10.1007/BFb0104029Google Scholar
  32. M. Sanchón, J. M. Urbano: Entropy solutions for the p(x)-Laplace equation, Trans. Amer. Math. Soc. 361 (2009) 6387–6405.https://doi.org/10.1090/S0002-9947-09-04399-2Google Scholar
  33. P. Wittbold, A. Zimmermann: Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and L1 -data, Nonlinear Anal. 72 (2010) 2990–3008.https://doi.org/10.1016/j.na.2009.11.041Google Scholar
  34. E. Zeidler: Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators, Springer, New York (1990).https://doi.org/10.1007/978-1-4612-0981-2Google Scholar
  35. C. Zhang, S. Zhou: Entropy and renormalized solutions for the p(x) Laplacian equation with measure data, Bull. Aust. Math. Soc. 82 (2010) 459–479.https://doi.org/10.1017/S0004972710000432Google Scholar
  36. C. Zhang, S. Zhou: Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L1 data, J. Differ. Equations 248 (2010) 1376–1400.https://doi.org/10.1016/j.jde.2009.11.024Google Scholar
  37. V. V. Zhikov: On the density of smooth functions in Sobolev-Orlicz spaces, Zap. Nauchn. Semin. POMI 226 (2004) 67–81.Google Scholar

Suggested citation

B. Lv, F. Li, W. Zou. “Existence and Uniqueness of Renormalized Solutions to Some Nonlinear Elliptic Equations with Variable Exponents and Measure Data.” Journal of Convex Analysis 21 (2014), No. 2, 317–338. https://doi.org/10.68381/jca21018

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.