Concentrating solutions for an anisotropic planar elliptic Neumann problem with Hardy-Hénon weight and large exponent (Q2090618)

This paper studies the following anisotropic elliptic Neumann problem with Hardy-Hénon weight \begin{align*} &-\nabla(a(x) \nabla u)+ a(x)u=a(x)|x-q|^{2\alpha}u^p,\ u>0,\ \text{ in }\Omega,\\ &\frac{\partial u}{\partial \nu}=0,\ \text{ on }\partial\Omega, \end{align*} where \(\nu\) denotes the outer unit normal vector to \(\partial \Omega\), \(q\in \Omega\), \(\alpha \in (-1,+\infty)\setminus\mathbb{N}\), \(p>1\) is a large exponent and \(a(x)\) is a positive smooth function. The main result is that if \(q\in \Omega\) is a strict local maximum point of \(a(x)\), there exists a family of positive solutions with arbitrarily many interior spikes accumulating to \(q\); while, if \(q\in \partial \Omega\) is a strict local maximum point of \(a(x)\) and satisfies \(\langle \nabla a(q),\nu(q)\rangle=0\), the problem has a family of positive solutions with arbitrarily many mixed interior and boundary spikes accumulating to \(q\). In particular, it is proved that concentration at singular source \(q\) is always possible whether \(q \in \bar\Omega\) is an isolated local maximum point of \(a(x)\) or not. The main argument is a finite dimensional reduction.