Abstract
We study a nonlinear equation in the half-space with a Hardy potential, specifically,
where Δp stands for the p-Laplacian operator defined by Δpu = div(∣Δu∣p−2Δu), p > 1, θ > −p, and T is a half-space {x1 > 0}. When λ > Θ (where Θ is the Hardy constant), we show that under suitable conditions on f and θ, the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as x1 → 0+, and the symmetric property of the positive solution are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balinsky A A, Evans W D, Lewis R T. The Analysis and Geometry of Hardy’s Inequality. Switzerland: Springer International Publishing, 2015: 77–134
Bandle C, Marcus M, Moroz V. Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential. Israel J Math, 2017, 222: 487–514
Bandle C, Moroz V, Reichel W. Boundary blowup type sub-solutions to semilinear elliptic equations with Hardy potential. J London Math Soc, 2008, 77: 503–523
Chen Y J, Wang M X. Boundary blow-up solutions for p-Laplacian elliptic equations of logistic type. Proc Roy Soc Edingburg Sect A, 2012, 142: 691–714
Du Y H, Guo Z M. Boundary blow-up solutions and their applications in quasilinear elliptic equations. J D’Analyse Math, 2003, 89: 277–302
Du Y H, Guo Z M. Symmetry for elliptic equations in a half-space without strong maximum principle. Proc Roy Soc Edingburg Sect A, 2004, 13: 259–269
Du Y H, Wei L. Boundary behavior of positive solutions to nonlinear elliptic equations with Hardy potential. J London Math Soc, 2015, 91: 731–749
García-Melián J, Sabina de Lis J. Maximum and comparison principles for operators involving the p-Laplacian. J Math Anal Appl, 1998, 218: 49–65
Guo Z, Ma L. Asymptotic behavior of positive solutions of some quasilinear elliptic problems. J London Math Sco, 2007, 76: 419–437
Jia H, Li G. Multiplicity and concentration behaviour of positive solutions for Schrödinger-Kirchhoff type equations involving the p-Laplacian in ℝN. Acta Math Sci, 2018, 38B(2): 391–418.
Li G, Yang T. The existence of a nontrivial weak solution to a double critical problem involving a fractional Laplacian in ℝN with a Hardy term. Acta Math Sci, 2020, 40B(6): 1808–1830
Marcus M, Mizel V J, Pinchover Y. On the best constant for Hardy’s inequality in Rn. Trans Amer Math Soc, 1998, 350: 3237–3255
Marcus M, Shafrir I. An eigenvalue problem related to Hardy’s Lp inequality. Ann Scuola Norm Sup Pisa Cl Sci, 2000, 29(4): 581–604
Vazquez J L. A strong maximum principle for some quasilinear elliptic equations. Appl Amath Optim, 1984, 12: 191–202
Wang M X. Nonlinear Elliptic Equations. Beijing: Science Press, 2010
Wei L. Asymptotic behavior and symmetry of positive solutions to nonlinear elliptic equations in a half-space. Proc Roy Soc Edinburgh Sect A, 2016, 146: 1243–1263
Youssfi A, Mahmoud G. On singular equations involving fractional Laplacian. Acta Math Sci, 2020, 40B(5): 1289–1315
Zeng X, Wan J. The existence of solution for p-Laplace equation involving resonant term. Acta Math Sci, 2014, 34A(2): 227–233
Author information
Authors and Affiliations
Corresponding author
Additional information
L. Wei was supported by NSFC (11871250). Y.M. Zhang was supported by NSFC (11771127, 12171379) and the Fundamental Research Funds for the Central Universities (WUT: 2020IB011, 2020IB017, 2020IB019).
Rights and permissions
About this article
Cite this article
Chen, Y., Wei, L. & Zhang, Y. The Asymptotic Behavior and Symmetry of Positive Solutions to p-Laplacian Equations in a Half-Space. Acta Math Sci 42, 2149–2164 (2022). https://doi.org/10.1007/s10473-022-0524-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-022-0524-y