Multiple boundary bubbling phenomenon of solutions to a Neumann problem (Q1027098)

The authors are concerned with existence and asymptotic behavior of nontrivial solutions to the boundary value problem \(-\mathrm{div}(a(x)\nabla u)+a(x)u=0\) in \(\Omega\), \(\partial u/\partial \nu=\varepsilon e^{u}\) on \(\partial\Omega\). Here \(\Omega\) is a smooth bounded domain in the plane and \(\varepsilon>0\) is small. The main novelty of the paper is the presence of the smooth weight \(a(x)\) which satisfies \(0<a_1\leq a(x)\leq a_2<\infty\) in \(\Omega\). First, the authors establish a decomposition result for the regular part of the associated Green function to the above problem. This decomposition result is then used to construct nontrivial solutions with a cluster of multiple bubbles near a local maximum point of \(a(x)\) on the boundary.