Multiplicity of weak solutions to $p(\cdot)$-Laplacian problem with discontinuous Steklov boundary conditions

Abdolrahman Razani; Farzaneh Safari

doi:10.5269/bspm.63947

Abdolrahman Razani Imam Khomeini International University
Farzaneh Safari

Abstract

The focus of this paper is to investigate the existence and multiplicity of weak solutions for an elliptic equation including discontinuous Steklov boundary conditions. The approach taken in this study involves the utilization of variational methods.

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References

Aberqi, A., Bennouna, J., Benslimane, O. and Ragusa, M. A., On p(z)-Laplacian system involving critical nonlinearities, J. Funct. Spaces 2022, no. 6685771, (2022). https://doi.org/10.1155/2022/6685771

Bonanno, G., A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75, 2992-3007, (2012).

Bonanno, G. and Marano, S. A., On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89, 1-18, (2010).

Bonanno, G., Relatin between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1, 205–220, (2012).

Bonanno, G., Candito, P. and D’Aguı, G. , Variational methods on finite dimensional Banach spaces and discrete problems, Adv. Nonlinear Stud. 14, no. 4, 915–939, (2014).

Canavati, J. and Minzoni, A. A., A discontinuous Steklov problem with an application to water waves, J. Math. Anal. Appl. 69, no. 2, 540-558, (1979).

Chammem, R., Ghanmi, A. and Sahbani, A., Existence and multiplicity of solutions for some Styklov problem involving p(x)-Laplacian operato, Appl. Anal. 101 , no. 7, 2401-2417, (2022).

Diening, L., Harjulehto, P., Hasto, P. and Rauzicka, M. , Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, Springer, Heidelberg, (2011).

Evans, L. C., Partial Differential Equations: Second Edition, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, (2010).

Fan, X. L. and Zhao, D., On the generalized Orlicz-Sobolev space W k,p(x)(Ω), J. Gansu Educ. College 12, no. 1, 1-6,(1986).

Karagiorgos, Y. and Yannakaris, N., A Neumann problem involving the p(x)-Laplacian with p = ∞ in a subdomain, Adv. Calc. Var. 9, no. 1, 65-76, (2016).

Karagiorgos, Y. and Yannakaris, N., A Neumann problem for the p(x)-Laplacian with p = 1 in a subdomain, J. Math. Anal. 454, no. 1, 412-428, (2017).

Khaled Ben, A., Existence results for Steklov problem involving the p(x)-Laplace operator, Complex Var. Elliptic Equ. 63, no. 12, 1675-1686, (2018).

Khaleghi, A. and Razani, A., Existence and multiplicity of solutions for p(x)-Laplacian problem with Steklov boundary condition, Bound. Value Probl. 2022, no. 39, 11 pages, (2022). https://doi.org/10.1186/s13661-022-01624-y

Khaleghi, A. and Safari, F., Multiplicity of weak solutions for the Steklov systems involving the Laplacian operator, Filomat, 38:21 (2024), 7541–7549. https://doi.org/10.2298/FIL2421541K

Kovacik, O. and Rakosnik, J., On spaces Lp(·) and W k,p(·), Czechoslov. Math. J. 41, no. 4, 592-618, (1991).

Ragusa, M. A., Razani, A. and Safari, F., Existence of radial solutions for a p(x)-Laplacian Dirichlet problem, Adv. Differential Equations no. 1, 1-14, (2021).

Ragusa, M. A., Razani, A. and Safari, F., Existence of positive radial solutions for a problem involving weighted Heisenberg p(x)-Laplacian operator, AIMS Mathematics 8, no.1, 404-422, (2023). https://doi.org/10.3934/math.2023019

Razani, A. and Safari, F., A (p(x), q(x)-Laplacian problem with the Steklov boundary conditions, Lobachevskii J. Math. 43, no. 12, 3616-3625, (2022). https://doi.org/10.1134/S1995080222150252

Razani, A. and Safari, F., Solutions to a (p1, · · · , pn)-Laplacian problem with Hardy potentials, J. Nonlinear Math. Phys. 30, no. 2, 413-427, (2023). https://doi.org/10.1007/s44198-022-00089-y

Razani, A. and Safari, F., An elliptic type inclusion problem on the Heisenberg Lie group, Math. Slovaca 73, no. 4, 957-968 (2023). https://doi.org/10.1515/ms-2023-0071

Razani, A., Safari, F. and Figueiredo, G. M., Existence and multiplicity of solutions for a weighted (p, q)-Laplacian

problem on the Heisenberg Lie groups, Bull. Belgian. Math. Soc. 30 (3), 281-296, (2023). https://doi.org/10.36045/j.bbms.220219

Razani, A. and Safari, F., Existence results to a Leray-Lions type problem on the Heisenberg Lie groups, Bound. Value Probl. 18, (2023). https://doi.org/10.1186/s13661-023-01704-7

Razani, A., Safari, F. and Soltani, T., Weak solutions for a system involving anisotropic (->p (·), ->q (·))-Laplacian operators, Iran J. Sci., (2024). https://doi.org/10.1007/s40995-024-01627-7

Safari, F. and Razani, A., Existence of positive radial solutions for Neumann problem on the Heisenberg group, Bound. Value Probl. 2020, no.88, (2020). https://doi.org/10.1186/s13661-020-01386-5

Safari, F. and Razani, A., Nonlinear nonhomogeneous Neumann problem on the Heisenberg group, Appl. Anal. 101, Issue 7, 2387-2400, (2022). https://doi.org/10.1080/00036811.2020.1807013

Safari, F., and Razani, A., Existence of radial solutions for a weighted p-biharmonic problem with Navier boundary condition on the Heisenberg group, Math. Slovaca 72, No. 3, 677-692, (2022). https://doi.org/10.1515/ms-2022-0046

Safari, F. and Razani, A., Radial solutions for a general form of a p-Laplace equation involving nonlinearity terms, Complex Var. Elliptic Equ. 68, no. 3, 361-371, (2023). https://doi.org/10.1080/17476933.2021.1991331