# American Institute of Mathematical Sciences

January  2020, 40(1): 133-151. doi: 10.3934/dcds.2020006

## Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space

 Department of Mathematics, West University of Timişoara, 4, Blvd. V. Pârvan 300223, Timişoara, Romania

* Corresponding author: Petru Jebelean

Received  November 2018 Published  October 2019

We deal with a multiparameter Dirichlet system having the form
 $\begin{equation*} \left\{ \begin{array}{ll} -\mathcal M(u) = \lambda_1f_1(u,v), & \hbox{in$\Omega$},\\ -\mathcal M(v) = \lambda_2f_2(u,v), & \hbox{in$\Omega$},\\ u|_{\partial\Omega} = 0 = v|_{\partial\Omega}, \end{array} \right. \end{equation*}$
where
 $\mathcal M$
stands for the mean curvature operator in Minkowski space
 $\mathcal M(u) = \mbox{div} \left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right),$
 $\Omega$
is a general bounded regular domain in
 $\mathbb{R}^N$
and the continuous functions
 $f_1,f_2$
satisfy some sign and quasi-monotonicity conditions. Among others, these type of nonlinearities, include the Lane-Emden ones. For such a system we show the existence of a hyperbola like curve which separates the first quadrant in two disjoint sets, an open one
 $\mathcal{O}_0$
and a closed one
 $\mathcal{F}$
, such that the system has zero or at least one strictly positive solution, according to
 $(\lambda_1, \lambda_2)\in \mathcal{O}_0$
or
 $(\lambda_1, \lambda_2)\in \mathcal{F}$
. Moreover, we show that inside of
 $\mathcal{F}$
there exists an infinite rectangle in which the parameters being, the system has at least two strictly positive solutions. Our approach relies on a lower and upper solutions method - which we develop here, together with topological degree type arguments. In a sense, our results extend to non-radial systems some recent existence/non-existence and multiplicity results obtained in the radial case.
Citation: Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006
##### References:

show all references

##### References:
 [1] Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271 [2] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 [3] Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1921-1933. doi: 10.3934/dcdss.2020150 [4] Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765 [5] Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012 [6] Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3443-3462. doi: 10.3934/dcds.2019142 [7] Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061 [8] Chunyan Ji, Yang Xue, Yong Li. Periodic solutions for SDEs through upper and lower solutions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4737-4754. doi: 10.3934/dcdsb.2020122 [9] João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217 [10] Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575 [11] Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014 [12] Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102 [13] Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147 [14] Rubén Figueroa, Rodrigo López Pouso, Jorge Rodríguez–López. Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 617-633. doi: 10.3934/dcdsb.2019257 [15] Alberto Cabada, João Fialho, Feliz Minhós. Non ordered lower and upper solutions to fourth order problems with functional boundary conditions. Conference Publications, 2011, 2011 (Special) : 209-218. doi: 10.3934/proc.2011.2011.209 [16] Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure and Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963 [17] Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89 [18] Yuxia Guo, Jianjun Nie. Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6873-6898. doi: 10.3934/dcds.2016099 [19] Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971 [20] Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180

2021 Impact Factor: 1.588