Positive solutions for BVPs with one-dimensional mean curvature operator (Q2923135)

In this paper, motivated by an open problem described by \textit{C. Bereanu} et al. [Proc. Am. Math. Soc. 137, No. 1, 161--169 (2009; Zbl 1161.35024)], the authors discuss, by using variational methods, the existence and the properties of solutions to systems of Dirichlet problems involving one dimensional mean curvature operator, \( -\left( \frac{x^{\prime }(t)}{\sqrt{1+(x^{\prime }(t))^{2}}}\right) =\frac{ \partial \widetilde{H}}{\partial x}(t,x(t),y(t),u(t),w(t)),\) \(-\left( \frac{ y^{\prime }(t)}{\sqrt{1+(y^{\prime }(t))^{2}}}\right) =\frac{\partial \widetilde{H}}{\partial y}(t,x(t),y(t),u(t,w(t)),\) \(x(0)=x(1)=0,\text{ } y(0)=y(1)=0\text{,}\) where \(\widetilde{H}\) satisfies certain properties. First, they show for each pair of functional parameters \((u,v)\) fixed in a certain space, that there exists at least one positive solution and then they study the continuous dependence (in some sense) of the solutions on functional parameters under only local conditions made on the nonlinearities.