Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction

Abstract

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85(3–4) (2013) 199–227) where the linear case $p = 2$ is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.

Keywords

p-Laplacian uniform elliptcity Poincaré inequality Krasnoselskii’s genus

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References

C.O.

Alves,

Julio,

S.A.

Corrêa and

Chipot, On a class of intermediate local-nonlocal elliptic problems, Topol. Methods Nonlinear Anal. 49(2) (2017), 497–509.

Chipot, On the asymptotic behaviour of some problems of the calculus of variations, J. Elliptic Parabol. Equ. 1 (2015), 307–323. doi:10.1007/BF03377383.

Chipot, Asymptotic Issues for Some Partial Differential Equations, Imperial College Press, London, 2016.

Chipot, On some elliptic problems in unbounded domains, Chinese Ann. Math. Ser. B 39(3) (2018), 597–606. doi:10.1007/s11401-018-0083-7.

Chipot, Asymptotic Issues for Some Partial Differential Equations, 2nd edn, World Scientific Publishing Company, 2024.

Chipot,

Dávila and

del Pino, On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains, J. Fixed Point Theory Appl. 19(1) (2017), 205–213. doi:10.1007/s11784-016-0349-1.

Chipot and

Mardare, On correctors for the Stokes problem in cylinders, in: Nonlinear Phenomena with Energy Dissipation, GAKUTO Internat. Ser. Math. Sci. Appl., Vol. 29, Gakkotosho, Tokyo, 2008, pp. 37–52.

Chipot and

Mardare, Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction, J. Math. Pures Appl. (9) 90(2) (2008), 133–159. doi:10.1016/j.matpur.2008.04.002.

Chipot and

Mardare, The Neumann problem in cylinders becoming unbounded in one direction, J. Math. Pures Appl. (9) 104(5) (2015), 921–941. doi:10.1016/j.matpur.2015.05.008.

10.

Chipot,

Mojsic and

Roy, On some variational problems set on domains tending to infinity, Discrete Contin. Dyn. Syst. 36(7) (2016), 3603–3621. doi:10.3934/dcds.2016.36.3603.

11.

Chipot and

Rougirel, On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions, Discrete Contin. Dyn. Syst. Ser. B 1(3) (2001), 319–338.

12.

Chipot and

Rougirel, Remarks on the asymptotic behaviour of the solution to parabolic problems in domains becoming unbounded, in: Proceedings of the Third World Congress of Nonlinear Analysts, Part 1, Catania, 2000, Vol. 47, 2001, pp. 3–11.

13.

Chipot and

Rougirel, On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded, Commun. Contemp. Math. 4(1) (2002), 15–44. doi:10.1142/S0219199702000555.

14.

Chipot and

Rougirel, On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded, Trans. Amer. Math. Soc. 360(7) (2008), 3579–3602. doi:10.1090/S0002-9947-08-04361-4.

15.

Chipot,

Roy and

Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity, Asymptot. Anal. 85(3–4) (2013), 199–227.

16.

Chipot and

Xie, On the asymptotic behaviour of the p-Laplace equation in cylinders becoming unbounded, in: Nonlinear Partial Differential Equations and Their Applications, GAKUTO Internat. Ser. Math. Sci. Appl., Vol. 20, Gakkotosho, Tokyo, 2004, pp. 16–27.

17.

Chipot and

Xie, Asymptotic behavior of nonlinear parabolic problems with periodic data, in: Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Vol. 63, Birkhäuser, Basel, 2005, pp. 147–156. doi:10.1007/3-7643-7384-9_17.

18.

Chipot and

Zube, On the asymptotic behaviour of the pure Neumann problem in cylinder-like domains and its applications, Asymptot. Anal. 108(3) (2018), 163–185.

19.

Chowdhury,

Csató,

Roy and

Sk, Study of fractional Poincaré inequalities on unbounded domains, Discrete Contin. Dyn. Syst. 41(6) (2021), 2993–3020. doi:10.3934/dcds.2020394.

20.

Chowdhury and

Roy, On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity, Commun. Contemp. Math. 19(5) (2017), 1650035, 21.

21.

Chowdhury and

Roy, On fractional Poincaré inequality for unbounded domains with finite ball conditions: Counter example, Nonlinear Anal. 228 (2023), 113189, 16.

22.

Drábek and

S.B.

Robinson, Resonance problems for the p-Laplacian, J. Funct. Anal. 169(1) (1999), 189–200. doi:10.1006/jfan.1999.3501.

23.

Esposito,

Roy and

Sk, On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded, Asymptot. Anal. 123(1–2) (2021), 79–94.

24.

Guesmia and

Harkat, Long time behaviour of parabolic equations in time-dependent growing domains, Asymptot. Anal. 108(4) (2018), 187–219.

25.

Kesavan, Topics in Functional Analysis and Applications, John Wiley & Sons, Inc., New York, 1989.

26.

Lê, Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64(5) (2006), 1057–1099. doi:10.1016/j.na.2005.05.056.

27.

Le Dret and

Mokrane, On problems in the calculus of variations in increasingly elongated domains, Chinese Ann. Math. Ser. B 39(2) (2018), 163–182. doi:10.1007/s11401-018-1058-4.

28.

Lindqvist, A nonlinear eigenvalue problem, in: Topics in Mathematical Analysis, Ser. Anal. Appl. Comput., Vol. 3, World Sci. Publ., Hackensack, NJ, 2008, pp. 175–203. doi:10.1142/9789812811066_0005.

29.

Lotoreichik and

Rohleder, Eigenvalue inequalities for the Laplacian with mixed boundary conditions, J. Differential Equations 263(1) (2017), 491–508. doi:10.1016/j.jde.2017.02.043.

30.

Prosenjit, Some results in asymptotic analysis and nonlocal problems, Thesis.

31.

Purbita, Anisotropic p-Laplace equations on long cylindrical domain, Opuscula Math. 44(2) (2024), 249–265. doi:10.7494/OpMath.2024.44.2.249.

32.

Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990.

33.

Yeressian, Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89(1–2) (2014), 21–35.