Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data (Q2362313)

The paper is devoted to the study of the existence and concentration behavior of boundary bubbling solutions to the problem \[ -\Delta v+v=0\text{ in }\Omega, \] \[ {\partial u}/{\partial \nu} =e^v-s\phi_1 -h(x)\text{ on }\partial\Omega, \] where \(\Omega\subset \mathbb R^2\) is a bounded domain with smooth boundary, \(\nu\) denotes the outer unit normal vector to \(\partial\Omega\), \(h\in C^{0,\alpha}(\partial\Omega)\), \(s>0\) is a large parameter and \(\phi_1\) is a positive first eigenfunction of the Steklov problem \[ -\Delta \phi+\phi=0\text{ in }\Omega, \] \[ {\partial \phi}/{\partial \nu} =\lambda \phi\text{ on }\partial\Omega. \] The authors prove the existence of a family of solutions of the problem with accumulation of arbitrarily many boundary bubbles around the maximum points of \(\phi_1\). Moreover, they construct solutions which exhibit a multiple boundary concentration behavior around the maximum points of \(\phi_1\) on the boundary, as \(s\to +\infty\).