Sign-changing concentration phenomena of an anisotropic sinh-Poisson type equation with a Hardy or Hénon term

We consider the following anisotropic sinh-Poisson tpye equation with a Hardy or Hénon term: $$-\mathrm{d}\mathrm{i}\mathrm{v}(a(x)\mathrm{\nabla }u)+a(x)u={\epsilon }^{2}a(x)\mathrm{\mid }x-q{\mathrm{\mid }}^{2\alpha }(e^u-e^{-u})\mathrm{i}\mathrm{n}\mathrm{\Omega },$$ $$\frac{\mathrm{\partial }u}{\mathrm{\partial }n}=0,\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },$$ where $\epsilon >0$, $q\in \bar{\mathrm{\Omega }}\subset {\mathbb{R}}^2$, $\alpha \in (-1,\mathrm{\infty })-\mathbb{N}$, $\mathrm{\Omega }\subset {\mathbb{R}}^2$ is a smooth bounded domain, $n$ is the unit outward normal vector of $\mathrm{\partial }\mathrm{\Omega }$ and $a(x)$ is a smooth positive function defined on $\bar{\mathrm{\Omega }}$. From finite dimensional reduction method, we proved that this problem has a sequence of sign-changing solutions with arbitrarily many interior spikes accumulating to $q$, provided $q\in \mathrm{\Omega }$ is a local maximizer of $a(x)$. However, if $q\in \mathrm{\partial }\mathrm{\Omega }$ is a strict local maximum point of $a(x)$ and satisfies $⟨\mathrm{\nabla }a(q),n⟩=0$, we proved that this problem has a family of sign-changing solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$. Under the same condition, we could also construct a sequence of blow-up solutions for the following problem $$-\mathrm{d}\mathrm{i}\mathrm{v}(a(x)\mathrm{\nabla }u)+a(x)u={\epsilon }^{2}a(x)\mathrm{\mid }x-q{\mathrm{\mid }}^{2\alpha }{e}^u\mathrm{i}\mathrm{n}\mathrm{\Omega },$$ $$\frac{\mathrm{\partial }u}{\mathrm{\partial }n}=0,\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$