EXISTENCE RESULTS FOR ANISOTROPIC FRACTIONAL (p1(x, .), p2(x, .))-KIRCHHOFF TYPE PROBLEMS

E. Azroul; A. Benkirane; N. T. Chung; M. Shimi

doi:10.11948/20200394

2021 Volume 11 Issue 5

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E. Azroul, A. Benkirane, N. T. Chung, M. Shimi. EXISTENCE RESULTS FOR ANISOTROPIC FRACTIONAL (p1(x, .), p2(x, .))-KIRCHHOFF TYPE PROBLEMS[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2363-2386. doi: 10.11948/20200394

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E. Azroul, A. Benkirane, N. T. Chung, M. Shimi. EXISTENCE RESULTS FOR ANISOTROPIC FRACTIONAL (p₁(x, .), p₂(x, .))-KIRCHHOFF TYPE PROBLEMS[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2363-2386. doi: 10.11948/20200394

EXISTENCE RESULTS FOR ANISOTROPIC FRACTIONAL (p₁(x, .), p₂(x, .))-KIRCHHOFF TYPE PROBLEMS

1.
Department of Mathematics, Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar el Mahraz, Laboratory of Mathematical Analysis and Applications, B.P. 1796 Fez-Atlas, 30003, Fez, Morocco
2.
Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam

Corresponding author: Email: mohammed.shimi2@usmba.ac.ma (M. Shimi)

Abstract

Abstract

In this paper, we investigate the existence and multiplicity of solutions for a class of fractional $ (p_1(x,.),p_2(x,.)) $-Kirchhoff type problems with Dirichlet boundary data of the following form

$ \left( \mathcal{P}_{M_i}^s\right) \; \; \left\{ \begin{array}{lllll} \sum\limits_{i = 1}^{2} M_i \left(\int_{Q}\frac{1}{p_i(x,y)}\frac{|u(x)-u(y)|^{p_i(x,y)}}{|x-y|^{N+sp_i(x,y)}}\; dxdy\right) \left( -\Delta\right)_{p_i(x,.)} ^{s}u(x)\\+\sum\limits_{i = 1}^{2}|u|^{\bar{p}_i(x)-2}u = f(x,u)\quad\rm{in}\; \; \Omega,\\ u = 0 \quad \quad\rm{in}\; \; \; \mathbb{R}^{N}\setminus\Omega. \end{array}\right. $

More precisely, by means of mountain pass theorem with Cerami condition, we show that the above problem has at least one nontrivial solution. Moreover, using Fountain theorem, we prove that $ \left( \mathcal{P}_{M_i}^s\right) $ possesses infinitely many (pairs) of solutions with unbounded energy.
- Kirchhoff type problems /
- fractional p(x, .)-Laplacian operator /
- mountain pass theorem /
- fountain theorem /
- cerami condition
MSC: 35R11, 47G20, 35S15, 35A15

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References

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