An equation that arises in studying ground state solutions to semilinear elliptic equations (Q1916644)

The author considers the equation \[ - u_{xx}- \Delta_z u= g(u),\quad (x, z)\in (- \ell, \ell)\times \mathbb{R}^N,\tag{\(*\)} \] where \(u\) satisfies periodic boundary conditions in \(x\). He studies properties of the mapping \(\ell\mapsto u^\ell\), where \(u^\ell\) is a ground state solution of \((*)\).