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.