Positive solutions for nonlinear singular boundary value problem of magnetic insulation (Q2764305)

The second-order differential system of the form NEWLINE\[NEWLINE \phi''(x) = j{{1+\phi(x)}\over{Q(\phi(x),a(x))}}, \qquad a''(x) = j{a(x)\over{Q(\phi(x),a(x))}}, NEWLINE\]NEWLINE with the parameter \(j\) and \(Q(\phi,a)= [(1+\phi)^2 - a^2 -1]^{{1\over 2}}\), which may serve as a model of the process of magnetic insulation in plasma, and more general systems are considered together with the two-point boundary conditions \(\phi(0)=0\), \(\phi(1)=\phi_1\), \(a(0)=0\), \(a(1)=a_1\). Under a number of additional assumptions, the boundary value problem under consideration is shown to possess a positive solution, for which purpose a version of the method of upper and lower solutions is used.