Super-critical boundary bubbling in a semilinear Neumann problem (Q1766517)

In the present study the authors considers the following problem \[ -\Delta u+ u= u^{{N+2\over N-2}+\varepsilon}\quad\text{in }\Omega,\qquad u> 0\quad\text{in }\Omega,\qquad {\partial u\over\partial\nu}= 0\quad\text{on }\partial\Omega,\tag{1} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) and \(N\geq 3\). The authors prove the existence of a one-spike solutions to (1) which concentrates around a topologically non-trivial critical point of the mean curvature of the boundary with positive value. They are able to construct solutions with just one maximum point for which multiple bubbling is present. They show, for example if \(\Omega\) is a ball, there exists a solution whose shape is that of a tower, constituted by supersolution of an arbitrary number of single-bubbles of different blow-up orders. This phenomenon actually takes place just provided that \(\Omega\) is symmetric with respect to the first \((N-1)\) variables, and \(0\in\partial\Omega\) is a point with positive mean curvature.