International Journal of Nonlinear Analysis and Applications
Existence of solutions for a quasilinear elliptic system with variable exponent
Document Type : Research Paper
Authors
University of Sidi Mohamed Ben Abdellah, Faculty of Sciences Dhar El Mehraz, B.P. 1796 Atlas, Fez, Morocco
Abstract
We consider the following quasilinear elliptic system in a Sobolev space with variable exponent:
\[-\text{div}(a(|Du|)Du)=f,\]
where $a$ is a $C^1$-function and $f\in W^{-1,p'(x)}(\Omega;\R^m)$. We use the theory of Young measures and weak monotonicity conditions to obtain the existence of solutions.
Keywords
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[9] J. M. Ball, A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transitions, Lecture Notes Phys. 344 (1989) 207–215.
[10] N. Bourbaki, Integration I (S. Berberian, Trans.), Springer-Verlag, Berlin, Heidelberg, 2004 (Chapters 1-6).
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[12] A. Cianchi and V.G. Maz’ya, Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems, J. Eur. Math. Soc. (JEMS). 16(3) (2014) 571–595.
[13] E. Dibenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115(5) (1993) 1107–1134.
[14] X. L. Fan and D. Zhao, On the Spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001) 424–446.
[15] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Amer. Math. Soc. Number 74, 1990.
[16] X. L. Fan and D. Zhao, On the generalized Orlicz-Sobolev space Wk,p(x) (Ω), J. Gansu Educ. College 12(1) (1998) 1–6.
[17] N. Hunger¨uhler, A refinement of Ball’s theorem on Young measures, New York J. Math. 3 (1997) 48–53.
[18] N. Hungerb¨uhler, Quasilinear elliptic systems in divergence form with weak monotonicity, New York J. Math. 5 (1999) 83–90.
[19] J. Kinuunen and S. Zhou, A local estimates for nonlinear equations with discontinuous coefficients, Commun Partial Diff. Eq. 24 (1999) 2043–2068.
[20] O. Kov´a˘cik and J. R´akosn´ık, On spaces Lp(x) (Ω) and Wk,p(x) (Ω), Czechoslovak Math. J. 41(116) (1991) 592–618.
[21] M. Misawa, Lq estimates of gradients for evolutional p-Laplacian systems, J. Diff. Eq. 219 (2005) 390-420.
[22] K. Yosida, Functional Analysis, Springer, Berlin, 1980.
[23] E. Zeidler, Nonlinear Functional Analysis and its Application, Volume I, Springer, 1986.
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Volume 12, Issue 2
November 2021Pages 205-217
- Receive Date: 02 August 2019
- Revise Date: 28 October 2019
- Accept Date: 29 October 2019