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We study the semilinear problem with the boundary reaction $$-\Delta u+u=0\text{in}\Omega ,\frac{\partial u}{\partial \nu }=\lambda f\left(u\right)\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$, $N\ge 2$, is a smooth bounded domain, $f:[0,\infty )\to (0,\infty )$ is a smooth, strictly positive, convex, increasing function which is superlinear at $\infty$, and $\lambda >0$ is a parameter. It is known that there exists an extremal parameter ${\lambda }^{*}>0$ such that a classical minimal solution exists for $\lambda <{\lambda }^{*}$, and there is no solution for $\lambda >{\lambda }^{*}$. Moreover, there is a unique weak solution $u^{*}$ corresponding to the parameter $\lambda ={\lambda }^{*}$. In this paper, we continue to study the spectral properties of $u^{*}$ and show...

This note is concerned with the recent paper "Non-topological N-vortex condensates for the self-dual Chern-Simons theory" by M. Nolasco. Modifying her arguments and statements, we show that the existence of "non-topological" multi-vortex condensates follows when the number of prescribed vortex points is greater than or equal to 2.