Solutions of a nonlinear boundary value problem with a large parameter (Q1304698)

The ordinary differential equation \[ u_{xx}+ u^p= 0,\quad x\in (-L, L), \] is considered together with the Dirichlet boundary conditions \(u(-L)= u_\ell\), \(u(L)= u_r\). Depending on \(p\), \(u_\ell\), \(u_r\), the existence or nonexistence, the number and structure of solutions are studied. Their limits as \(p\to\infty\) are described, which are piecewise linear functions.