We deal with the existence and localization of positive radial solutions for Dirichlet problems involving \(\phi\)-Laplacian operators in a ball. In particular, \(p\)-Laplacian and Minkowski-curvature equations are considered.
Our approach relies on fixed point index techniques, which work thanks to a Harnack-type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions.
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
compression–expansion, Dirichlet problem, fixed point index, Harnack-type inequality, mean cur-vature operator, Positive radial solution
R. Precup, J. Rodríguez-López, Positive radial solutions for Dirichlet problems via a Harnack-type inequality, Mathematical Methods in the Applied Sciences, 46 (2023) no. 2, pp. 2972-2985, https://doi.org/10.1002/mma.8682
Mathematical Methods in the Applied Sciences
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 Benedikt J, Girg P, Kotrla L, Takac P. Origin of thep-Laplacian and A. Missbach.Electron J Differ Equ. 2018;2018(16):1-17.
 Bartnik R, Simon L. Spacelike hypersurfaces with prescribed boundary values and mean curvature.Commun Math Phys. 1982;87:131-152.
 Ecker K. Mean curvature evolution of spacelike hypersurfaces, Proceedings of the Centre for Mathematics and its Applications.AustralianNat Univ. 1999;37:119-132.
 Athanassenas M, Clutterbuck J. A capillarity problem for compressible liquids.Pacific J Math. 2009;243:213-232.
 López-Gómez J, Omari P, Rivetti S. Positive solutions of a one-dimensional indefinite capillarity-type problem:A variational approach.J Differ Equ. 2017;262:2335-2392.
 Bereanu C, Jebelean P, Mawhin J. Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces.Proc Amer Math Soc. 2009;137(1):161-169.
 Bereanu C, Jebelean P, ̧Serban C. Dirichlet problems with mean curvature operator in Minkowski space. In: New Trends in Differential Equations, Control Theory and Optimization: Proceedings of the 8th Congress of Romanian Mathematicians; 2016:1-20.
 Bereanu C, Jebelean P, Torres PJ. Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space.J FunctAnal. 2013;264:270-287.
 Bereanu C, Jebelean P, Torres PJ. Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space.J Funct Anal. 2013;265:644-659.
 Coelho I, Corsato C, Rivetti S. Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol.Methods Nonlinear Anal. 2014;44(1):23-39.
 Corsato C, Obersnel F, Omari P, Rivetti S. Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space.J Math Anal Appl. 2013;405(1):227-239.
 García–Huidobro M, Manásevich R, Schmitt K. Positive radial solutions of quasilinear elliptic partial differential equations on a ball. Nonlinear Anal. 1999;35:175-190.
 He X. Multiple radial solutions for a class of quasilinear elliptic problems. Appl Math Lett. 2010;23(1):110-114.
 Ma R, Gao H, Lu Y. Global structure of radial positive solutions for a prescribed mean curvature problem in a ball. J Funct Anal.2016;270:2430-2455.
 Herlea D-R, Precup R. Existence, localization and multiplicity of positive solutions to -Laplace equations and systems. Taiwan J Math.2016;20:77-89.
 Precup R, Rodríguez-López J. Positive solutions for discontinuous problems with applications to -Laplacian equations. J Fixed PointTheory Appl. 2018;20:1-17.
 Precup R, Pucci P, Varga C. Energy-based localization and multiplicity of radially symmetric states for the stationary p-Laplace diffusion.Complex Var Elliptic Equ. 2020;65:1198-1209.
 Precup R. Moser–Harnack inequality, Krasnosel’skiı type fixed point theorems in cones and elliptic problems. Topol Methods Nonlin Anal.2012;40:301-313.