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
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[8] P. Harjulehto , P. H¨ast¨o, M. Koskenoja and S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal. 25 (2006), 205–222.
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[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.
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[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.
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Volume 14, Issue 1
January 2023Pages 3201-3210
- Receive Date: 03 November 2022
- Revise Date: 08 January 2023
- Accept Date: 13 January 2023