Multi-layer radial solutions for a supercritical Neumann problem (Q272260)

The paper under review deals with the Neumann problem NEWLINENEWLINE\[NEWLINE \begin{cases} -\Delta u+u=u^p & \text{in}\;B_1,\\ u>0, & \\ NEWLINE\partial_\nu u=0 & \text{on}\;\partial B_1, \end{cases} NEWLINE\]NEWLINE NEWLINEwhere \(B_1\) is the unit ball in \(\mathbb{R}^N\) with \(N\geq3\) and \(p>1\).NEWLINENEWLINEThe authors prove existence of multiple layer solutions as \(p\to\infty.\) These are radial solutions whose Laplacians weakly converge to measures concentrating at interior spheres, with a simple reflection rule.NEWLINENEWLINEThe machinery employed rely on a gluing technique, using a variant of the Nehari method, adapted to deal with Neumann problems instead of the standard Dirichlet ones.