O'zbekcha
English
Русский
RESEARCH AROUND ONE WEAK SOLUTION FOR SPECIAL OPERATOR IN VARIABLE EXPONENT SPACES
Maqola haqida umumiy ma'lumotlar
We prove the existence of one weak solution for the fourth-order problem involving the special operator in variable exponent spaces. The proof of our main result uses variational methods.
This-type problems are used to describe a large class of physical phenomena Such as micro-electro-mechanical systems, phase field models of multiphase Systems, thin film theory, thin plate theory, surface diffusion on Solids, interface dynamics, and also flow in Hele–Shaw cells. That is why many authors have looked for solutions of elliptic equations involving such operators.
[1] S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara., 52 (2006), 19–36, https://doi:10.1007/s11565-006-0002-9.
[2] S. N. Antontsev and S. I. Shmarev, A model porous medium equa tion withvariable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 60 (2005), 515–545, https:\doi: 10.1016/j.na.2004.09.026.
[3] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009 (2009), 1–20.
[4] M. M. Boureanu, Fourth-order problems with special type operators in variable exponent spaces, Discrete. Contin. Dyn. Syst. Ser. S., 12 (2019), No. 2, 231-243. https://doi:10.3934/dcdss.2019016.
[5] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383–1406, https://doi:10.1137/050624522.
[6] L. Diening, P. Harjulehto, P. Hästö and M. Ru̇žička, Lebesgue and Sobolev spaces with variable exponents, Springer-Verlag Berlin Heidelberg 2011.
[7] X. L. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424–446, https://doi.org/10.1006/jmaa.2000.7617.
[8] X. Fan, Q. Zhang and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306–317, https://doi.org/10.1016/j.jmaa.2003.11.020.
[9] K. Kefi, D. D. Repovs̆and K. Saoudi, On weak solutions for fourth order problems involving the special type operators, Math. Methods. Appl. Sci., 44 (2021), No. 17, 13060–13068.
[10] J. Leray and J. L. Lions, Quelques résultats de Vis̆ik sur les problem elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France., 93 (1965) 97–107.
[11] V. D. Rădulescu and D. D. Repovs̆, Partial differential equations with variable exponents: variational methods and qualitative analysis, Chapman and Hall/CRC. 2015.
[12] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401–410.
[13] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory. Lect. Notes Math., 1748 (2000), 16–38.
[14] A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal. T.M.A., 69 (2008), 3629–3636.
[15] V. V. Zhikov, Lavrentiev phenomenon and homogenization for some variational problems, C R Acad Sci Paris Sér I Math., 316 (1993), No. 5, 435–439.
Elbegi, Y. . (2023). RESEARCH AROUND ONE WEAK SOLUTION FOR SPECIAL OPERATOR IN VARIABLE EXPONENT SPACES. Academic Research in Educational Sciences, 4(7), 34–41. https://doi.org/
Elbegi, Yalda. “RESEARCH AROUND ONE WEAK SOLUTION FOR SPECIAL OPERATOR IN VARIABLE EXPONENT SPACES.” Academic Research in Educational Sciences, vol. 7, no. 4, 2023, pp. 34–41, https://doi.org/.
Elbegi, . 2023. RESEARCH AROUND ONE WEAK SOLUTION FOR SPECIAL OPERATOR IN VARIABLE EXPONENT SPACES. Academic Research in Educational Sciences. 7(4), pp.34–41.