Three weak solutions for a class of fourth order p(x)-Kirchhoff type problem with Leray-Lions operators
DOI :
https://doi.org/10.5269/bspm.62936Résumé
In this work, we study the multiplicity of a weak solution for a fourth order p(x)-Kirchhoff type problem involvingthe Leray-Lions type operators with no flux boundary condition. By using variational approach and critical point
theory, we determine an open interval of parameters for which our problem admits at least three distinct weak solutions.
Références
1. Arosio, A., Panizzi, S., On the well-posedness of the Kirchhoff string, Trans. Amer. Math. soc. 348, 305-330, (1996).
2. Bonanno, G., Some remarks on a three critical points theorem, Nonlinear Anal. TMA. 54, 651-665, (2003).
3. Bonanno, G., Marano, S. A., On the structure of the critical set of nondifferentiable functions with a weak compactness condition, Appl. Anal. 89, 1-10, (2010).
4. Boureanu, M.M., Fourth order problems with Leray-Lions type operators in variable exponent spaces, Discrete Contin. Dyn. Syst. Ser. S. 12, 231-243, (2019).
5. Boureanu, M.M., Radulescu, V.D., Repovs, D., On a p(.)-biharmonic problem with no-flux boundary condition, Comput. Math. App. 72, 2505-2515, (2016).
6. Cruz-Uribe, D.V., Fiorenza, A., Variable Lebesgue spaces. Foundations and harmonic analysis. Applied and Numerical Harmonic Analysis, Brikhauser/Springer, Heidelberg (2013).
7. Dreher, M., The kirchhoff equation for the p-Laplacian. Rend. Semin. Mat. Univ. Politec. Torino. 64, 217-238, (2006).
8. Edmunds, D. E., Rakosnık, J., Sobolev embeddings with variable exponent, studia Math. 143, 267-293,(2000).
9. El Amrouss, A.R., Ourraoui, A.: Existence of solutions for a boundary problem involving p(x)-biharmonic operator, Bol. Soc. Parana. Mat. (3) 31, 179-192,(2013).
10. Fan, X.L., Han, X., Existence and multiplicity of solutions for p(x)-Laplacian equations in RN . Nonlinear Anal. 59, 173-188, (2004).
11. Fan, X.L., Zhao, D., On the spaces Lp(x) (Ω) and W m,p(x) (Ω) , J. Math. Anal. appl. 263, 424-446, (2001).
12. Kefi, K., Radulescu, V.D., Small perturbations of nonlocal biharmonic problems with variable exponent and competing nonlinearities, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29, 439-463, (2018).
13. Kefi, K., Repovs, D.D., Saoudi, K., On weak solutions for fourth-order problems involving the Leray-Lions type operators, Math. Meth. Appl. Sci. (2021). https://doi.org/10.1002/mma.7606
14. Kefi, K., Irzi, N., Al-Shomrani, M.M., Repovs, D.D., On the fourth-order Leray-Lions problem with indefinite weight and nonstandard growth conditions, Bulletin of Mathematical Sciences. (2021). https://doi.org/10.1142/S1664360721500089
15. Al-Shomrani, M.M., Salah, M.B.M., Ghanmi, A., Kefi, K., Existence Results for Nonlinear Elliptic Equations with Leray–Lions Operators in Sobolev Spaces with Variable Exponents, Math Notes 110, 830-841, (2021). https://doi.org/10.1134/S0001434621110201
16. Kirchhoff, G., Mechanik, Teubner, Leipzig. (1883).
17. Kovacik, O., Rakosnık, J., On spaces Lp(x) and Wk,p(x) , Czechoslovak Math. J. 41, 592-618, (1991).
18. Leray, J., Lions, J.L., Quelques resultats de Visik sur les probl`emes elliptiques non lineaires par les methodes de Minty-Browder, Bulletin de la Societe Mathematique de France. 93, 97-107, (1965).
19. Lions, J.L., On some questions in boundary value problems of mathematical physics, in Proceedings of International Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro 1977. Math. Stud. (Penha and Medeiros, Eds), North Holland. 30, 284-346, (1978).
20. Matei, P.: Nemytskij operators in Lebesgue spaces with a variable exponent, Rom. J. Math. Comput. Sci. 3, 109-118, (2013).
21. Radulescu, V.D., Repovs, D.D., Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL. (2015).
22. Soualhine, K., Filali, M., Talbi, M., Tsouli, N., A critical p(x)-biharmonic Kirchhoff type problem with indefinite weight under no flux boundary condition, Bol. Soc. Mat. Mex. 28, 22,(2022). https://doi.org/10.1007/s40590-022-00419-6
23. Talbi, M., Filali, M., Soualhine, K., Tsouli, N., On a p(x)-biharmonic Kirchhoff type problem with indefinite weight and no flux boundary condition, Collect. Math. (2021). https://doi.org/10.1007/s13348-021-00316-7
24. Tsouli, N., Haddaoui, M., Hssini EL M., Multiple Solutions for a Critical p(x)-Kirchhoff Type Equations, Bol. Soc. Paran. Mat. 38 197-211, (2020).
25. Zang, A., Fu, Y., Interpolation inequalities for derivatives in variable exponent Lebesgue Sobolev spaces, Nonlinear Anal. 69, 3629-3636, (2008).
2. Bonanno, G., Some remarks on a three critical points theorem, Nonlinear Anal. TMA. 54, 651-665, (2003).
3. Bonanno, G., Marano, S. A., On the structure of the critical set of nondifferentiable functions with a weak compactness condition, Appl. Anal. 89, 1-10, (2010).
4. Boureanu, M.M., Fourth order problems with Leray-Lions type operators in variable exponent spaces, Discrete Contin. Dyn. Syst. Ser. S. 12, 231-243, (2019).
5. Boureanu, M.M., Radulescu, V.D., Repovs, D., On a p(.)-biharmonic problem with no-flux boundary condition, Comput. Math. App. 72, 2505-2515, (2016).
6. Cruz-Uribe, D.V., Fiorenza, A., Variable Lebesgue spaces. Foundations and harmonic analysis. Applied and Numerical Harmonic Analysis, Brikhauser/Springer, Heidelberg (2013).
7. Dreher, M., The kirchhoff equation for the p-Laplacian. Rend. Semin. Mat. Univ. Politec. Torino. 64, 217-238, (2006).
8. Edmunds, D. E., Rakosnık, J., Sobolev embeddings with variable exponent, studia Math. 143, 267-293,(2000).
9. El Amrouss, A.R., Ourraoui, A.: Existence of solutions for a boundary problem involving p(x)-biharmonic operator, Bol. Soc. Parana. Mat. (3) 31, 179-192,(2013).
10. Fan, X.L., Han, X., Existence and multiplicity of solutions for p(x)-Laplacian equations in RN . Nonlinear Anal. 59, 173-188, (2004).
11. Fan, X.L., Zhao, D., On the spaces Lp(x) (Ω) and W m,p(x) (Ω) , J. Math. Anal. appl. 263, 424-446, (2001).
12. Kefi, K., Radulescu, V.D., Small perturbations of nonlocal biharmonic problems with variable exponent and competing nonlinearities, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29, 439-463, (2018).
13. Kefi, K., Repovs, D.D., Saoudi, K., On weak solutions for fourth-order problems involving the Leray-Lions type operators, Math. Meth. Appl. Sci. (2021). https://doi.org/10.1002/mma.7606
14. Kefi, K., Irzi, N., Al-Shomrani, M.M., Repovs, D.D., On the fourth-order Leray-Lions problem with indefinite weight and nonstandard growth conditions, Bulletin of Mathematical Sciences. (2021). https://doi.org/10.1142/S1664360721500089
15. Al-Shomrani, M.M., Salah, M.B.M., Ghanmi, A., Kefi, K., Existence Results for Nonlinear Elliptic Equations with Leray–Lions Operators in Sobolev Spaces with Variable Exponents, Math Notes 110, 830-841, (2021). https://doi.org/10.1134/S0001434621110201
16. Kirchhoff, G., Mechanik, Teubner, Leipzig. (1883).
17. Kovacik, O., Rakosnık, J., On spaces Lp(x) and Wk,p(x) , Czechoslovak Math. J. 41, 592-618, (1991).
18. Leray, J., Lions, J.L., Quelques resultats de Visik sur les probl`emes elliptiques non lineaires par les methodes de Minty-Browder, Bulletin de la Societe Mathematique de France. 93, 97-107, (1965).
19. Lions, J.L., On some questions in boundary value problems of mathematical physics, in Proceedings of International Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro 1977. Math. Stud. (Penha and Medeiros, Eds), North Holland. 30, 284-346, (1978).
20. Matei, P.: Nemytskij operators in Lebesgue spaces with a variable exponent, Rom. J. Math. Comput. Sci. 3, 109-118, (2013).
21. Radulescu, V.D., Repovs, D.D., Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL. (2015).
22. Soualhine, K., Filali, M., Talbi, M., Tsouli, N., A critical p(x)-biharmonic Kirchhoff type problem with indefinite weight under no flux boundary condition, Bol. Soc. Mat. Mex. 28, 22,(2022). https://doi.org/10.1007/s40590-022-00419-6
23. Talbi, M., Filali, M., Soualhine, K., Tsouli, N., On a p(x)-biharmonic Kirchhoff type problem with indefinite weight and no flux boundary condition, Collect. Math. (2021). https://doi.org/10.1007/s13348-021-00316-7
24. Tsouli, N., Haddaoui, M., Hssini EL M., Multiple Solutions for a Critical p(x)-Kirchhoff Type Equations, Bol. Soc. Paran. Mat. 38 197-211, (2020).
25. Zang, A., Fu, Y., Interpolation inequalities for derivatives in variable exponent Lebesgue Sobolev spaces, Nonlinear Anal. 69, 3629-3636, (2008).
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2024-05-03
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Research Articles
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