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.