Radial non-potential Dirichlet systems with mean curvature operator in Minkowski space

Abstract

We deal with a multiparameter Dirichlet system having the form

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathcal {M}}(\text{ u })+\lambda _1\mu _1(|x|)f_1(\text{ u },\text{ v })=0 &{} { \text{ in } {\mathcal {B}}(R)},\\ {\mathcal {M}}(\text{ v })+\lambda _2\mu _2(|x|)f_2(\text{ u },\text{ v })=0 &{} { \text{ in } {\mathcal {B}}(R)},\\ \text{ u }|_{\partial {\mathcal {B}}(R)}=0=\text{ v }|_{\partial {\mathcal {B}}(R),} \end{array} \right. \end{aligned}$$

where \({\mathcal {M}}\) stands for the mean curvature operator in Minkowski space, \({\mathcal {B}}(R)\) is an open ball of radius R in \({\mathbb {R}}^N,\) the parameters \(\lambda _1,\lambda _2\) are positive, the functions \(\mu _1,\; \mu _2:[0,R]\rightarrow [0,\infty )\) are continuous and positive and the continuous functions \(f_1,f_2\) satisfy some sign, growth and monotonicity conditions. Among others, these type of nonlinearities, include the Lane-Emden ones. For this system we show that there exists a continuous curve \(\varGamma \) splitting the first quadrant into two disjoint unbounded, open sets \({\mathcal {O}}_1\) and \({\mathcal {O}}_2\) such that the system has zero, at least one or at least two positive radial solutions according to \((\lambda _1, \lambda _2)\in {\mathcal {O}}_1,\) \((\lambda _1, \lambda _2)\in \varGamma \) or \((\lambda _1, \lambda _2)\in {\mathcal {O}}_2,\) respectively. The set \({\mathcal {O}}_1\) is adjacent to the coordinates axes \(0 \lambda _1\) and \(0 \lambda _2\) and the curve \(\varGamma \) approaches asymptotically to two lines parallel to the axes \(0 \lambda _1\) and \(0 \lambda _2\). Actually, this result extends to more general radial systems the recent existence/non-existence and multiplicity result obtained in the case of Lane-Emden systems.

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Correspondence to Daniela Gurban.

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Gurban, D. Radial non-potential Dirichlet systems with mean curvature operator in Minkowski space. Positivity 25, 109–119 (2021). https://doi.org/10.1007/s11117-020-00751-z

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Keywords

  • Minkowski curvature operator
  • Multiparameter system
  • Positive solution
  • Non-existence/existence
  • Multiplicity

Mathematics Subject Classification

  • 35J66
  • 34B15
  • 34B18