International Journal of Nonlinear Analysis and Applications

Generalized $p$-Laplacian systems with lower order terms

Document Type : Review articles

Authors

University of Sidi Mohamed Ben Abdellah, Faculty of Sciences Dhar El Mehraz, B.P. 1796 Atlas, Fez, Morocco

Abstract

This work is devoted to studying the existence of solutions to systems of $p$-Laplacian type. We prove the existence of at least one weak solution, under some assumptions, by applying Galerkin's approximation and the theory of Young measures.

Keywords

[1] A. Abassi, A. El Hachimi and A. Jamea, Entropy solutions to nonlinear Neumann problems with L1-data, Int. J. Math. Stat. 2 (2008) 4–17.
[2] E. Azroul and F. Balaadich, Weak solutions for generalized p-Laplacian systems via Young measures, Moroccan J. of Pure Appl. Anal. 4(2) (2018) 77–84.
[3] E. Azroul, F. Balaadich, Quasilinear elliptic systems in perturbed form, Int. J. Nonlinear Anal. Appl. 10(2) (2019) 255–266.
[4] E. Azroul and F. Balaadich, A weak solution to quasilinear elliptic problems with perturbed gradient, Rend. Circ. Mat. Palermo Series 2 70 (2020) 151-166.
[5] E. Azroul and F. Balaadich, On strongly quasilinear elliptic systems with weak monotonicity, J. Appl. Anal. 27(1) (2021) 153–162.
[6] F. Balaadich and E. Azroul, On a class of quasilinear elliptic systems, Acta Sci. Math. (Szeged) 87 (2021) 141-152.
[7] F. Balaadich and E. Azroul, Elliptic systems of p-Laplacian type, Tamkang J. Math. 53 (2022).
[8] J.M. Ball, A version of the fundamental theorem for Young measures, Rascle M., Serre D., Slemrod M. (eds) PDEs and Continuum Models of Phase Transitions. Lecture Notes in Physics, vol 344. Springer, Berlin, Heidelberg. 344 (1989) 207–215.
[9] N. Bourbaki, Integration I (S. Berberian, Trans.), Springer-Verlag, Berlin, Heidelberg, 2004.
[10] D. Breit, A. Cianchi, L. Diening, T. Kuusi and S. Schwarzaker, The p-Laplace system with right-hand side in divergence form: inner and up to the boundary pointwise estimates, Nonlinear Anal. Theory, Meth. Appl. 115 (2017) 200–212.
[11] J. Chabrowski, Degenerate elliptic equation involving a subcritical Sobolev exponent, Portugal. Math. 53 (1996) 167–177.
[12] F. Crispo, C-R. Grisanti and P. Maremont, On the high regularity of solutions to the p-Laplacian boundary value problem in exterior domains, Ann. Math. Pure Appl. 195 (2016) 821–834.
[13] J.I. Diaz and F. de Thelin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal. 25(4) (1994) 1085–1111.
[14] G. Dolzmann, N. Hungerbuhler, S. Muller, The p-harmonic system with measure-valued right-hand side, Ann. Inst. Henri Poincare, 14(3) (1997) 353–364.
[15] L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS, 1990.
[16] N. Hungerbuhler, Quasilinear elliptic systems in divergence form with weak monotonicity, New York J. Math. 5 (1999) 90.
[17] T. Iwaniec, Projections onto gradients fields and Lp-estimates for degenerated elliptic operators, Studia Math. 75 (3) (1983) 293–312.
[18] O. Kavian, Introduction a la Theorie des Points Critiques: et Applications aux Problemes Elliptiques, SpringerVerlag, 1993.
[19] P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth, J. Diff. Equ. 221 (2006) 412–443.
[20] S. EH. Miri, Existence of solutions to quasilinear elliptic problems with nonlinearity and absorption-reaction gradient term, Elect. J. Diff. Equ. 32 (2014) 1–12
[21] P. Pucci and R. Servadei, On weak solutions for p-Laplacian equations with weights, Discrete Cont. Dyn. Syst. 2007 (2007) 1–10.
[22] M. Struwe, Variational Methods, Second Edition, Springer Verlag Berlin, Heidelberg, New York, 1996.
[23] H. Beirao da Veiga and F. Crispo, On the global W2,q regularity for nonlinear N-systems of the p-Laplacian type in n space variables, Nonlinear Anal. 75 (2012) 4346–4354.
[24] L. Wei, H. Zhou, Research on the existence of solution of equation involving p-Laplacian operator, Appl. Math. Chinese Univ. Ser. B 21 (2) (2006) 191–202.
[25] L. Wei and R.P. Agarwal, Existence of solutions to nonlinear Neumann boundary value problems with generalized p-Laplacian operator, Comput. Math. Appl. 56 (2008) 530–541.
[26] E. Zeidler, Nonlinear functional analysis and its application I, Springer, 1986.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 45-55
  • Receive Date: 20 June 2020
  • Revise Date: 01 February 2021
  • Accept Date: 04 March 2021