Positive solutions of second order systems of boundary value problems
The authors consider the boundary value problem: \(x''(t)+ \lambda a(t)f(x(t),y(t))= 0\), \(y''(t)+ \lambda b(t)g(x(t),y(t))= 0\), \(x(0)= x(1)= y(0)= y(1)= 0\), where \(f\) and \(g\) are continuous vector functions, componentwise nonnegative and increasing in each variable, \(a\) and \(b\) are continuous and nonnegative diagonal matrices. In these assumptions, they determine a set of values of \(\lambda>0\) in order that the problem has a nonnegative solution \((x,y)\), proving that it is not trivial when \(f(0,0)= g(0,0)= 0\) and \(f\) and \(g\) are superlinear. The results apply, for instance, when \(f(x,y)= y^ p\), \(g(x,y)= x^ q\) \((p,q>1)\) or \(f(x,y)= e^ y\), \(g(x,y)= e^ x\). The method of proof is topological.
scientific article; zbMATH DE number 562695