Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane-Emden type nonlinearities
type: Item
instance of: scholarly article
title: Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane-Emden type nonlinearities
zbMATH Open document ID: 1421.35086
author: Daniela Gurban
author: P. Jebelean
published in: Journal of Differential Equations
publication date: 2019-03-26T00:00:00Z
review text: The authors of this article study the existence and multiplicity of positive solutions for non-variational radial systems of the form \[ \mathcal{M}(u)+g_1(|x|,u,v)=0 \text{ in }\Omega \] \[ \mathcal{M}(v)+g_1(|x|,u,v)=0 \text{ in }\Omega, \] with Dirichlet boundary conditions; that is, $u=0=v$ on $\partial \Omega$. Here, $\Omega$ is the ball with radius $R$ centered at the origin in $\mathbb{R}^N$ with $N\geq 2$ and the functions $g_1,g_2:[0,R]\times [0,\infty)^2\rightarrow [0,\infty]$ are continuous. The mean curvature opeartor $\mathcal{M}$ in Minkowski space is given by \[ \mathcal{M}(w)=\mathrm{div}\left(\frac{\nabla w}{\sqrt{1-|\nabla w|^2}}\right). \] At first, the existence of positive solutions for a class of differential systems containing a singular $\varphi$-Laplacian is established using the fixed point index in combination with the lower and upper solution method. Then, it is proven that there exists for a system with Lane-Embden type nonlinearities in a superlinear case a continuous curve $\Gamma$ separating the first quadrant into two disjoint unbounded sets. Additionally, it is shown that the system on these sets can either have at least one or two radial positive solutions.
reviewed by: Andreas Kleefeld
Mathematics Subject Classification ID: 35J47
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B09
zbMATH DE Number: 7041838
zbMATH Keywords: Minkowski curvature operator
zbMATH Keywords: Lane-Emden nonlinearities
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: positive solutions
MaRDI profile type: Publication
full work available at URL: j.jde.2018.10.030
OpenAlex ID: W2899371802
cites work: Spacelike hypersurfaces with prescribed boundary values and mean curvature
cites work: The Dirichlet problem with mean curvature operator in Minkowski space -- a variational approach
cites work: Corrigendum to: ``The Dirichlet problem with mean curvature operator in Minkowski space -- a variational approach''
cites work: Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space
cites work: Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space
cites work: Some remarks on a system of quasilinear elliptic equations
cites work: The homogeneous Hénon-Lane-Emden system
cites work: The Hénon-Lane-Emden system: a sharp nonexistence result
cites work: Multiplicity of positive solutions for a \((p_1,p_2)\)-Laplacian system and its applications
cites work: Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces
cites work: Existence of positive solutions for a nonvariational quasilinear elliptic system
cites work: Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball
cites work: The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz-Minkowski space
cites work: On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space
cites work: Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space
cites work: A variational approach to subquadratic perturbations of elliptic systems
cites work: Existence of positive radial solutions for a semilinear elliptic system without variational structure
cites work: Nonlinear elliptic systems
cites work: Q5748278
cites work: Existence and nonexistence of positive singular solutions for a class of semilinear elliptic systems
cites work: Boundary blow-up solutions to elliptic systems of competitive type
cites work: H-surfaces in Lorentzian manifolds
cites work: Q4421936
cites work: Positive radial solutions for systems with mean curvature operator in Minkowski space
cites work: Nontrivial solutions for potential systems involving the mean curvature operator in Minkowski space
cites work: On Lane-Emden type systems
cites work: Existence of multiple positive radial solutions for a semilinear elliptic system on an unbounded domain.
cites work: On positive solutions of the Dirichlet problem involving the extrinsic mean curvature operator
cites work: Global structure of radial positive solutions for a prescribed mean curvature problem in a ball
cites work: Non-existence of positive solutions of Lane-Emden systems
cites work: Entire spacelike hypersurfaces of constant mean curvature in Minkowski space
cites work: Q4296478
cites work: A priori estimates for a semilinear elliptic system without variational structure and their applications
DOI: 10.1016/J.JDE.2018.10.030
description: scientific article; zbMATH DE number 7041838