Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane-Emden type nonlinearities
The authors of this article study the existence and multiplicity of positive solutions for non-variational radial systems of the form \[ \mathcal{M}(u)+g_1(|x|,u,v)=0 \text{ in }\Omega \] \[ \mathcal{M}(v)+g_1(|x|,u,v)=0 \text{ in }\Omega, \] with Dirichlet boundary conditions; that is, $u=0=v$ on $\partial \Omega$. Here, $\Omega$ is the ball with radius $R$ centered at the origin in $\mathbb{R}^N$ with $N\geq 2$ and the functions $g_1,g_2:[0,R]\times [0,\infty)^2\rightarrow [0,\infty]$ are continuous. The mean curvature opeartor $\mathcal{M}$ in Minkowski space is given by \[ \mathcal{M}(w)=\mathrm{div}\left(\frac{\nabla w}{\sqrt{1-|\nabla w|^2}}\right). \] At first, the existence of positive solutions for a class of differential systems containing a singular $\varphi$-Laplacian is established using the fixed point index in combination with the lower and upper solution method. Then, it is proven that there exists for a system with Lane-Embden type nonlinearities in a superlinear case a continuous curve $\Gamma$ separating the first quadrant into two disjoint unbounded sets. Additionally, it is shown that the system on these sets can either have at least one or two radial positive solutions.
scientific article; zbMATH DE number 7041838
scientific article