We consider the following anisotropic sinh-Poisson type equation with a Hardy or Hénon term: 0.1 \( \left \{ \textstyle\begin{array}{l@{\quad}l} -\mathrm{div}(a(x)\nabla u)+ a(x)u=\varepsilon ^{2}a(x)|x-q|^{2\alpha}(e^{u}-e^{-u}) &\text{in $\Omega $,} \\ \frac{\partial u}{\partial n}=0, &\text{on $\Omega $,} \end{array}\displaystyle \right . \) where $\varepsilon >0$ , $q\in \bar{\Omega}\subset \mathbb{R}^{2}$ , $\alpha \in (-1,\infty )\backslash \mathbb{N}$ , $\Omega \subset \mathbb{R}^{2}$ is a smooth bounded domain, n is the unit outward normal vector of ∂Ω, and anisotropic coefficient $a(x)$ is a smooth positive function defined on Ω̄. From the finite-dimensional reduction method, we proved that the problem (0.1) has a sequence of sign-changing solutions with arbitrarily many interior spikes accumulating to q, provided $q\in \Omega $ is a strict local maximizer of $a(x)$ . However, if $q\in \partial \Omega $ is a strict local maximum point of $a(x)$ and satisfies $\langle \nabla a(q),n \rangle =0$ , we proved that (0.1) 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 to the following problem \( \left \{ \textstyle\begin{array}{l@{\quad}l} -\mathrm{div}(a(x)\nabla u)+ a(x)u=\varepsilon ^{2}a(x)|x-q|^{2\alpha}e^{u} &\text{in $\Omega $,} \\ \frac{\partial u}{\partial n}=0, &\text{on $\partial \Omega $.} \end{array}\displaystyle \right . \)
