On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows
The authors consider a parametrized boundary value problem with an integral constrained of the form \[ U''+ AU= U^2-{1\over\pi} \int^\pi_0 U^2(x)\,dx,\;x\in (0,\pi),\;U'(0)= U'(\pi)= 0,\;U'(x)> 0,\;\int^\pi_0 U(x)\,dx= 0, \] where \(A\in\mathbb{R}\) is a parameter. As the main result they show that all solutions \((A,U)\) to the above problem can be written in the form \[ \{(A(k), U(x,A(k)):0< k< 1\}, \] where \[ \begin{gathered} A(k)= {4K(k)\over\pi^2} (3E(k)+ (k^2- 2)K(k)),\\ U(x,A(k))= {6k^2K(k)^2\over \pi^2} c^2n\Biggl({K(k)\over\pi} x,k\Biggr)+ {6K(k)\over\pi^2} [E(k)- (1- k^2) K(k)].\end{gathered} \] The functions \(K\) and \(E\) are the well-known elliptic integrals given by \[ K(k)= \int^{\pi/2}_0 {d\varphi\over \sqrt{1-k^2\sin^2\varphi}}\quad\text{and}\quad E(k)= \int^{\pi/2}_0 \sqrt{1- k^2\sin^2\varphi} d\varphi,\;k\in [0,1). \]
scientific article; zbMATH DE number 2053146