Opuscula Math. 45, no. 4 (2025), 509-521
https://doi.org/10.7494/OpMath.2025.45.4.509
Opuscula Mathematica
Asymptotic behavior of the solutions of operators that are sum of pseudo p-Laplace type
Abstract. The article investigates a Poisson-type problem for operators that are finite sum of pseudo \(p\)-Laplace-type operators within long cylindrical domains. It establishes that the rate of convergence is exponential, which is considered optimal. In addition, the study analyzes the asymptotic behavior of the related energy functional. This research contributes to a deeper understanding of the mathematical properties and asymptotic analysis of solutions in this context.
Keywords: pseudo \(p\)-Laplace equation, cylindrical domains, asymptotic analysis.
Mathematics Subject Classification: 35B27, 46E35, 49J52.
- A. Alvino, G. Trombetti, J.I. Diaz, P.L. Lions, Elliptic equations and Steiner symmetrization, Comm. Pure Appl. Math. 49 (1996), 217-236. https://doi.org/10.1002/(sici)1097-0312(199603)49:3<217::aid-cpa1>3.0.co;2-g
- A. Brada, Comportement asymptotique de solutions d'équations elliptiques semi-linéares dans un cylindre, Asymptotic Anal. 10 (1995), 335-366. https://doi.org/10.3233/asy-1995-10402
- F. Brock, J.I. Díaz, A. Ferone, D. Gómez-Castro, A. Mercaldo, Steiner symmetrization for anisotropic quasilinear equations via partial discretization, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 38 (2021), 347-368. https://doi.org/10.1016/j.anihpc.2020.07.005
- M. Chipot, On the asymptotic behaviour of some problems of the calculus of variations, J. Elliptic Parabol. Equ. 1 (2015), 307-323. https://doi.org/10.1007/bf03377383
- M. Chipot, Asymptotic behaviour of operators sum of \(p\)-Laplacians, Discrete Contin. Dyn. Syst. Ser. S 17 (2024), 917-929. https://doi.org/10.3934/dcdss.2023117
- M. Chipot, Asymptotic behaviour of some anisotropic problems, Asymptot. Anal. 139 (2024), 217-243. https://doi.org/10.3233/asy-241906
- M.M. Chipot, Asymptotic Issues for Some Partial Differential Equations, 2nd ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2024. https://doi.org/10.1142/13765
- M. Chipot, W. Hackbusch, S. Sauter, A. Veit, Numerical approximation of Poisson problems in long domains, Vietnam J. Math. 50 (2022), 375-393. https://doi.org/10.1007/s10013-021-00512-9
- M. Chipot, A. Mojsic, P. Roy, On some variational problems set on domains tending to infinity, Discrete Contin. Dyn. Syst. 36 (2016) 7, 3603-3621. https://doi.org/10.3934/dcds.2016.36.3603
- M. Chipot, P. Roy, I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity, Asymptot. Anal. 85 (2013), 199-227. https://doi.org/10.3233/asy-131182
- J.I. Díaz, A. Ferone, A. Mercaldo, Anisotropic partial symmetrization for some quasilinear equations in comparison with a \(p\)-Laplace realization on some of the coordinates, J. Math. Anal. Appl. 548 (2025), Paper no. 129370. https://doi.org/10.1016/j.jmaa.2025.129370
- J.I. Díaz, S. González, New results on the Burgers and the linear heat equations in unbounded domains, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 99 (2005), 219-225.
- J.I. Diaz, O.A. Oleinik, Nonlinear elliptic boundary value problems in unbounded domains and the asymptotic behaviour of its solutions, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 787-792.
- L. Djilali, A. Rougirel, Galerkin method for time fractional diffusion equations, J. Elliptic Parabol. Equ. 4 (2018), 349-368. https://doi.org/10.1007/s41808-018-0022-5
- Y. Dolak, C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math. 66 (2005) 1, 286-308. https://doi.org/10.1137/040612841
- L.C. Evans, Partial Differential Equations, 2nd ed., volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. https://doi.org/10.1090/gsm/019
- J. Haškovec, C. Schmeiser, A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems, Monatsh. Math. 158 (2009), 71-79. https://doi.org/10.1007/s00605-008-0059-x
- P. Jana, Anisotropic \(p\)-Laplace equations on long cylindrical domain, Opuscula Math. 44 (2024), 249-265. https://doi.org/10.7494/opmath.2024.44.2.249
- J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York-Heidelberg, 1972.
- R. Rawat, H. Roy, P. Roy, Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction, Asymptot. Anal. 139 (2024), 245-277. https://doi.org/10.3233/asy-241907
- Purbita Jana
https://orcid.org/0000-0002-2817-7735- Madras School of Economics, 269Q+2CX, Gandhi Mandapam Road, Surya Nagar, Kotturpuram, Chennai, Tamil Nadu 600025, India
- Communicated by J.I. Díaz.
- Received: 2025-04-25.
- Revised: 2025-06-30.
- Accepted: 2025-07-06.
- Published online: 2025-07-16.
