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Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, westudy the following anisotropic elliptic Neumann problem with Hardy-H\'{e}nonweight $$ \begin{cases} -\nabla(a(x)\nablau)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\,\Omega,\\[2mm] \frac{\partial u}{\partial\nu}=0\,\,\qquad\quad\qquad\qquad\qquad \qquad\qquad\qquad\qquad \,\ \ \,\,\,\,\textrm{on}\,\,\, \partial\Omega, \end{cases} $$ where $\nu$ denotes the outerunit normal vector to $\partial\Omega$, $q\in\overline{\Omega}$,$\alpha\in(-1,+\infty)\setminus\mathbb{N}$, $p>1$ is a large exponent and$a(x)$ is a positive smooth function. We investigate the effect of theinteraction between anisotropic coefficient $a(x)$ and singular source $q$ onthe existence of concentrating solutions. We show that if $q\in\Omega$ is astrict local maximum point of $a(x)$, there exists a family of positivesolutions 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$\langle\nabla a(q),\,\nu(q)\rangle=0$, such a problem has a family of positivesolutions with arbitrarily many mixed interior and boundary spikes accumulatingto $q$. In particular, we find that concentration at singular source $q$ isalways possible whether $q\in\overline{\Omega}$ is an isolated local maximumpoint of $a(x)$ or not.

Concentrating solutions for an anisotropic planar elliptic Neumann problem with Hardy-H\'{e}non weight and large exponent