Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane–Emden type nonlinearities

Gurban, Daniela; Jebelean, Petru

doi:10.1016/j.jde.2018.10.030

Cited by 14 publications

(7 citation statements)

References 26 publications

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“…However, to the authors' best knowledge, the study of the Dirichlet problem of a quasilinear differential system with mean curvature operator M seems to be in its early stages, we refer the reader to [10-13, 16, 17, 21] and the references therein. For instance, Gurban et al [11] investigated the following two-parameter problem…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On a power-type coupled system with mean curvature operator in Minkowski space

Zhao

Miao

2021

Bound Value Probl

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We study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space $$ \textstyle\begin{cases} \mathcal{M}(u)+ v^{\alpha }=0\quad \text{in } B, \\ \mathcal{M}(v)+ u^{\beta }=0\quad \text{in } B, \\ u|_{\partial B}=v|_{\partial B}=0, \end{cases} $$ { M ( u ) + v α = 0 in B , M ( v ) + u β = 0 in B , u | ∂ B = v | ∂ B = 0 , where $\mathcal{M}(w)=\operatorname{div} ( \frac{\nabla w}{\sqrt{1-|\nabla w|^{2}}} )$ M ( w ) = div ( ∇ w 1 − | ∇ w | 2 ) and B is a unit ball in $\mathbb{R}^{N} (N\geq 2)$ R N ( N ≥ 2 ) . We use the index theory of fixed points for completely continuous operators to obtain the existence, nonexistence and uniqueness results of positive radial solutions under some corresponding assumptions on α, β.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On a power-type coupled system with mean curvature operator in Minkowski space

Zhao

Miao

2021

Bound Value Probl

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“…This allows us to depict multiplicity and localization informations about solutions. We note that our lower and upper solutions method -which is different from the one in [16,17] used for radial systems, as well as the multiplicity and localization results we obtain for problem (1) (Propositions 3.3 -3.5) are inspired by the corresponding ones proved for a single equation in [13,14].…”

Section: Introductionmentioning

confidence: 95%

“…Using Theorem 2.3 in [17], we infer that there exist λ * 1 , λ * 2 > 0 such that, for each λ 1 > λ * 1 and λ 2 > λ * 2 , problem (25) has a radial solution (u * , v * )…”

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confidence: 93%

“…[3,5,6,13,14,15,20]), while at our best knowledge, for systems with such an operator the study was recently initiated in [18]. Also, results for systems having a radial structure in a ball were obtained in [4,16,17]. So, using a variational approach, in the case of one parameter systems of type…”

Section: Introductionmentioning

confidence: 99%

“…This result extends the corresponding one obtained in [7] in the case of a single equation. Then, in the recent paper [17] were considered non-potential radial systems having the form…”

Section: Introductionmentioning

confidence: 99%

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

Gurban¹,

Jebelean²,

Şerban³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We deal with a multiparameter Dirichlet system having the formwhere M stands for the mean curvature operator in Minkowski spaceΩ is a general bounded regular domain in 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 O 0 and a closed one F , such that the system has zero or at least one strictly positive solution, according to (λ 1 , λ 2 ) ∈ O 0 or (λ 1 , λ 2 ) ∈ F . Moreover, we show that inside of 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.2010 Mathematics Subject Classification. Primary: 35J66, 34B16; Secondary: 34B18.

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