Multi-layer solutions for an elliptic problem. (Q1414011)

This paper considers the Neumann problem on the unit ball in \(\mathbb{R}^N\) for the equation \[ -\epsilon^2\Delta u=u(u-a(| x| ))(1-u), \] where \(\epsilon >0\) is a small parameter and \(a(r)\) is a continuous function satisfying \(0<a(r)<1\). It is proved that for any \(k\in \mathbb{N}^*\) and for \(\epsilon\) small enough, this problem has a radial solution with \(k\) layers near a point \(r_0\) with \(a(r_0)=\frac{1}{2}\) and \(a'(r_0)\neq 0\). These solutions are different from the global minimizer of the corresponding variational problem. The proof is variational and uses a reduction method to transform the problem into a finite dimension one.