Topological degree methods for a Neumann problem governed by nonlinear elliptic equation

Abbassi, Adil; Allalou, Chakir; Kassidi, Abderrazak

doi:10.2478/mjpaa-2020-0018

Abstract

In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation

$- d i v a (x, u, \nabla u) = b (x) {| u |}^{p - 2} u + λ H (x, u, \nabla u),$ - div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right),

where Ω is a bounded smooth domain of 𝕉^N.

References

[1] A. Abbassi, E. Azroul, A. Barbara, Degenerate p(x)-elliptic equation with second membre in L¹. Advances in Science, Technology and Engineering Systems Journal. Vol. 2, No. 5 (2017), 45-54.10.25046/aj020509
Search in Google Scholar Back to article
[2] A. Abbassi, C. Allalou and A. Kassidi, Existence of weak solutions for nonlinear p-elliptic problem by topological degree, Nonlinear Dyn. Syst. Theory, 20(3) (2020) 229-241.10.1007/s41808-021-00098-w
Search in Google Scholar Back to article
[3] S. Aizicovici, N.S. Papageorgiou, V. Staicu, Nodal solutions for nonlinear nonhomogeneous Neumann equations, Topol. Methods Nonlinear Anal. 43 (2014) 421-438.10.12775/TMNA.2014.025
Search in Google Scholar Back to article
[4] Y. Akdim, E. Azroul and A. Benkirane, Existence of solutions for quasilinear degenerate elliptic equations. Electronic Journal of Differential Equations (EJDE), 2001, vol. 2001, p. Paper No. 71, 19.
Search in Google Scholar Back to article
[5] C. Atkinson and C. R. Champion, Some boundary value problems for the equation ∇.(| ∇ ϕ |^N∇ ϕ) = 0, Quart. J. lllcch. Appl. Math. 37 (1984), 401-419.10.1093/qjmam/37.3.401
Search in Google Scholar Back to article
[6] C. Atkinson and C. W. Jones, Similarity solutions in some non-linear diffusion problems and in boundary-layer flow of a pseudo plastic fluid, Quart. J. Mech. Appl. Malh. 27 (1974), 193-211.10.1093/qjmam/27.2.193
Search in Google Scholar Back to article
[7] J. Baranger and K. Najib, Numerical analysis for quasi-Newtonian flow obeying the power Iaw or the Carreau law, Numer. Math. 58 (1990), 35-49.10.1007/BF01385609
Search in Google Scholar Back to article
[8] J. Berkovits, Extension of the LeraySchauder degree for abstract Hammerstein type mappings. Journal of Differential Equations, 2007, vol. 234, no 1, p. 289-310.10.1016/j.jde.2006.11.012
Search in Google Scholar Back to article
[9] G. Bonanno, G. D’Agui and A. Sciammetta. Nonlinear elliptic equations involving the p-laplacian with mixed Dirichlet-Neumann boundary conditions. Opuscula Mathematica, 2019, vol. 39, no 2.10.7494/OpMath.2019.39.2.159
Search in Google Scholar Back to article
[10] F. E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 1-39.10.1090/S0273-0979-1983-15153-4
Search in Google Scholar Back to article
[11] Y.J. Cho, and Y. Q. Chen. Topological degree theory and applications. Chapman and Hall/CRC, 2006.10.1201/9781420011487
Search in Google Scholar Back to article
[12] P. Drabek, A. Kufner and F. Nicolosi, Non Linear Elliptic Equations, Singular and Degenerated Cases. University of West Bohemia, 1996.
Search in Google Scholar Back to article
[13] P. Drabek, A. Kufner and V. Mustonen, Pseudo-monotonicity and degenerate or singular elliptic operators, Bull. Austral. Math. Soc. Vol. 58 (1998), 213-221.10.1017/S0004972700032184
Search in Google Scholar Back to article
[14] M. Filippakis, N.S. Papageorgiou, Nodal solutions for Neumann problems with a nonhomogeneous differential operator, Funkcial. Ekvac. 56 (2013) 63-79.10.1619/fesi.56.63
Search in Google Scholar Back to article
[15] J. P. Garca Azorero, and I. Peral Alonso, Existence and nonuniqueness for the p-Laplacian, Communications in Partial Differential Equations 12.12 (1987): 126-202.10.1080/03605308708820534
Search in Google Scholar Back to article
[16] L. Gasiński, N.S. Papageorgiou, Multiple solutions for a class of nonlinear Neumann eigenvalue problems, Commun. Pure Appl. Anal. 13 (2014) 1491-151210.3934/cpaa.2014.13.1491
Search in Google Scholar Back to article
[17] M. A. Hammou, E. Azroul and B. Lahmi, Existence of solutions for p(x)-Laplacian Dirichlet problem by topological degree, Bulletin of the Transilvania University of Brasov ffl Vol 11(60), No. 2-2018, Series III: Mathematics, Informatics, Physics, 29-38.
Search in Google Scholar Back to article
[18] M. A. Hammou, E. Azroul and B. Lahmi, Topological degree methods for a Strongly nonlinear p(x)-elliptic problem. Revista Colombiana de Matemticas 53.1 (2019): 27-39.10.15446/recolma.v53n1.81036
Search in Google Scholar Back to article
[19] A. Hirn, Approximation of the p-Stokes equations with equal-order finite elements, J. Math. Fluid Mech. 15 (2013) 65-88.10.1007/s00021-012-0095-0
Search in Google Scholar Back to article
[20] S.C. Hu, N.S. Papageorgiou, Nonlinear Neumann problems with indefinite potential and concave terms, Commun. Pure Appl. Anal. 14 (2015) 2561 – 2616.10.3934/cpaa.2015.14.2561
Search in Google Scholar Back to article
[21] T. He, Z. A. Yao and Z. Sun, Multiple and nodal solutions for parametric Neumann problems with nonhomogeneous differential operator and critical growth. Journal of Mathematical Analysis and Applications, 2017, vol. 449, no 2, p. 1133-1151.10.1016/j.jmaa.2016.12.020
Search in Google Scholar Back to article
[22] B. Kawohl, On a family of torsional creep problems, J. R. Angew. Math. 410 (1990) 1-22.10.1515/crll.1990.410.1
Search in Google Scholar Back to article
[23] I.S. Kim, A topological degree and applications to elliptic problems with discontinuous nonlinearity, J. Nonlinear Sci. Appl. 10, 612-624 (2017).10.22436/jnsa.010.02.24
Search in Google Scholar Back to article
[24] J. Leray, J. Schauder, Topologie et quations fonctionnelles, Ann. Sci. Ec. Norm., 51 (1934), 45-7810.24033/asens.836
Search in Google Scholar Back to article
[25] S. Liu, Existence of solutions to a superlinear p-Laplacian equation, Electron. J. Differential Equations 66.6 (2001).
Search in Google Scholar Back to article
[26] Z. Liu, D. Motreanu, and S. Zeng. Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient. Calculus of Variations and Partial Differential Equations 58.1 (2019): 28.10.1007/s00526-018-1472-1
Search in Google Scholar Back to article
[27] S.E. Manouni, N.S. Papageorgiou, P. Winkert, Parametric nonlinear nonhomogeneous Neumann equations involving a nonhomogeneous differential operator, Monatsh. Math. 177 (2015) 203-233.10.1007/s00605-014-0649-8
Search in Google Scholar Back to article
[28] D. Motreanu, V. V. Motreanu and N. Papageorgiou, Topological and variational methods with applications to nonlinear boundary value problems. Vol. 1. No. 7. New York: Springer, 2014.10.1007/978-1-4614-9323-5
Search in Google Scholar Back to article
[29] J. R. Philip, N-diffusion, Aust. J. Phiys. 14 (1961), 1-13.10.1071/PH610001
Search in Google Scholar Back to article
[30] M. Sabouri and M. Dehghan. A hk mortar spectral element method for the p-Laplacian equation. Computers and Mathematics with Applications 76.7 (2018): 1803-1826.10.1016/j.camwa.2018.07.031
Search in Google Scholar Back to article
[31] I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, Translated from the 1990 Russian original by Dan D. Pascali, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, (1994).10.1090/mmono/139
Search in Google Scholar Back to article
[32] I. V. Skrypnik, Nonlinear elliptic equations of higher order, (Russian) Gamoqeneb. Math. Inst. Sem. Mosen. Anotacie, 7 (1973), 51-52.
Search in Google Scholar Back to article
[33] E. Zeider, Nonlinear Functional Analysis and its Applications, II\B : Nonlinear Monotone Operators, Springer, New York, 1990.
Search in Google Scholar Back to article

Topological degree methods for a Neumann problem governed by nonlinear elliptic equation

Abstract

Paradigm

My account