International Journal of Nonlinear Analysis and Applications

Existence result for double phase problem involving the $(p(x),q(x))$-Laplacian-like operators

Document Type : Review articles

Authors

Laboratory LMACS, FST of Beni-Mellal, Sultan Moulay Slimane University, Morocco

Abstract

The paper study the existence of at least one weak solutions for Dirichlet boundary value problem involving the $\big(p(x),q(x)\big)$-Laplacian-like operators of the following form:
\begin{equation*}
\displaystyle\left\{\begin{array}{ll}
\displaystyle-\Delta^{l}_{p(x)}-\Delta^{l}_{q(x)}=\lambda g(x, u, \nabla u) & \mathrm{i}\mathrm{n}\ \Omega,\\\\
u=0 & \mathrm{o}\mathrm{n}\ \partial\Omega,
\end{array}\right.
\end{equation*}
where $\Delta^{l}_{r(x)} $ is the $r(x)$-Laplacian-like operators, $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $\lambda$ is a real parameter and $g$ is Carath'eodory function satisfies the assumption of growth. The existence is proved by using Berkovits' topological degree.

Keywords

[1] A. Abbassi, C. Allalou and A. Kassidi, Existence results for some nonlinear elliptic equations via topological degree methods, J. Elliptic Parabol Equ. 7 (2021), 121–136.
[2] A. Aberqi, O. Benslimane, M. Elmassoudi and M.A. Ragusa, Nonnegative solution of a class of double phase problems with logarithmic nonlinearity, Boundary Value Prob. 2022 (2022), no. 1, 1–57.
[3] E. Acerbi and G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 156 (2001), 121–140.
[4] C. Allalou and K. Hilal, Weak solution to p (x)-Kirchoff type problems under no-flux boundary condition by topological degree, Bol. Soc. Paranaense Mat. 41 (2023), 1–12.
[5] J. Berkovits, Extension of the Leray-Schauder degree for abstract Hammerstein type mappings, J. Differ. Equ. 234 (2007), 289–310.
[6] F. Behboudi and A. Razani, Two weak solutions for a singular (p, q)-Laplacian problem, Filomat 33 (2019), no. 11, 3399–3407.
[7] X.L. Fan and D. Zhao, On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001), 424–446.
[8] P. Harjulehto, P. Hasto, M. Koskenoja and S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal. 25 (2006), 205–222.
[9] S. Heidari and A. Razani, Iinfinitely many solutions for (p (x), q (x))-Laplacian-Like systems, Commun. Korean Math. Soc. 36 (2021), no. 1, 51–62.
[10] I.H. Kim and Y.H. Kim, Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math. 147 (2015), no. 1-2, 169–191.
[11] I.S. Kim and S.J. Hong, A topological degree for operators of generalized (S+) type, Fixed Point Theory Appl. 2015 (2015), no. 1, 1–16.
[12] O. Kovacik and J. Rakosnık, On spaces Lp(x) and W1,p(x) , Czech. Math. J. 41 (1991), no. 4, 592–618.
[13] M. Mahshi and A. Razani, A weak solution for a (p(x), q(x))-Laplacian elliptic problem with a singular term, Boundary Value Prob. 2021 (2021), 80.
[14] W.M. Ni and J. Serrin, Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo (2) Suppl. 8 (1985), 171–185.
[15] W.M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Conveg. Lincei. 77 (1986), 231–257.
[16] M.E. Ouaarabi, A. Abbassi and C. Allalou, Existence result for a Dirichlet problem governed by nonlinear degenerate elliptic equation in weighted Sobolev spaces, J Elliptic Parabol Equ. 7 (2021), no. 1, 221–242.
[17] M.E. Ouaarabi, A. Abbassi and C. Allalou, On the Dirichlet problem for some nonlinear degenerated elliptic equations with weight, 7th Int. Conf. Optim. Appl., 2021, pp. 1–6.
[18] M.E. Ouaarabi, A. Abbassi and C. Allalou, Existence result for a general nonlinear degenerate elliptic problems with measure datum in weighted Sobolev spaces, Int. J. Optim. Appl. 1 (2021), no. 2, 1–9.
[19] K.R. Rajagopal and M. R ˙uzicka, Mathematical modeling of electrorheological materials, Contin. Mech. Thermodyn. 13 (2001), no. 1, 59–78.
[20] M.A. Ragusa, A. Razani and F. Safari, Existence of radial solutions for a p(x)-Laplacian Dirichlet problem, Adv. Differ. Equ. 2021 (2021), 215.
[21] A. Razani, Two weak solutions for fully nonlinear Kirchhoff-type problem, Filomat 35 (2021), no. 10, 3267–3278.
[22] M. Manuela Rodrigues, Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators, Mediterr J Math. 9 (2012), 211–223.
[23] M. Ruzicka, Electrorheological fuids: Modeling and mathematical theory, Lecture Notes in Mathematics, Berlin Springer Verlag, 2000.
[24] S.G. Samko, Density of C∞ 0 (R N ) in the generalized Sobolev spaces W m,p(x) (R N ), Doklady Math. 60 (1999), no. 3, 382–385.
[25] S. Shokooh and A.Neirameh, Existence results of infinitely many weak solutions for p(x)-Laplacian-like operators, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 78 (2016), no. 4, 95–104.
[26] C. Vetro, Weak solutions to Dirichlet boundary value problem driven by p(x)-Laplacian-like operator, Electron. J. Qual. Theory Differ. Equ. 2017 (2017), no. 98, 1–10.
[27] S. Yacini, C. Allalou, K. Hilal and A.Kassidi, Weak solutions to Kirchhoff type problems via topological degree, Adv. Math. Mod. Appl. 6 (2021), no. 3, 309–321.
[28] E. Zeidler, Nonlinear functional analysis and its applications, II/B: Nonlinear monotone operators, Springer-Verlag, New York, 1990.
[29] Q.M. Zhou, On the superlinear problems involving p(x)-Laplacian-like operators without AR-condition. Nonlinear Anal. Real World Appl. 21 (2015), 161–169.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 1
January 2023
Pages 3201-3210
  • Receive Date: 03 November 2022
  • Revise Date: 08 January 2023
  • Accept Date: 13 January 2023