MULTIPLICITY OF WEAK SOLUTIONS FOR A (P(X), Q(X))-KIRCHHOFF EQUATION WITH NEUMANN BOUNDARY CONDITIONS

A. Ahmed; Mohamed Saad Bouh Elemine Vall

doi:10.11948/20230449

2024 Volume 14 Issue 4

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A. Ahmed, Mohamed Saad Bouh Elemine Vall. MULTIPLICITY OF WEAK SOLUTIONS FOR A (P(X), Q(X))-KIRCHHOFF EQUATION WITH NEUMANN BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2441-2465. doi: 10.11948/20230449

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A. Ahmed, Mohamed Saad Bouh Elemine Vall. MULTIPLICITY OF WEAK SOLUTIONS FOR A (P(X), Q(X))-KIRCHHOFF EQUATION WITH NEUMANN BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2441-2465. doi: 10.11948/20230449

MULTIPLICITY OF WEAK SOLUTIONS FOR A (P(X), Q(X))-KIRCHHOFF EQUATION WITH NEUMANN BOUNDARY CONDITIONS

A. Ahmed^1, ,,
Mohamed Saad Bouh Elemine Vall^2,

1.
Mathematics and Computer Sciences Department, Faculty of Science and Technology, University of Nouakchott, Nouakchott, Mauritania
2.
Department of Industrial Engineering and Applied Mathematics, Professional University Institute, University of Nouakchott, Nouakchott, Mauritania

Author Bio: Email: saad2012bouh@gmail.com(M. S. B. Elemine Vall)
Corresponding author: Email: ahmedmath2001@gmail.com(A. Ahmed)

Abstract

Abstract

The aim of this study is to investigate the existence of infinitely many weak solutions for the (p(x), q(x))-Kirchhoff Neumann problem described by the following equation:

$ \left\{\begin{array}{lc}-\left(a_1+a_2 \int_{\Omega} \frac{1}{p(x)}|\nabla u|^{p(x)} d x\right) \Delta_{p(\cdot)} u & \\ -\left(b_1+b_2 \int_{\Omega} \frac{1}{q(x)}|\nabla u|^{q(x)} d x\right) \Delta_{q(\cdot)} u & \\ +\lambda(x)\left(|u|^{p(x)-2} u+|u|^{q(x)-2} u\right)=f_1(x, u)+f_2(x, u) & \text { in } \Omega, \\ \frac{\partial u}{\partial \nu}=0 & \text { on } \partial \Omega .\end{array}\right. $

By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.
- Nonlinear elliptic equations /
- weak solutions to PDEs /
- Ricceri's variational principle /
- double phase problems /
- Musielak-Orlicz-Sobolev spaces
MSC: 35J60, 35D30, 35J20

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References

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