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In the past few decades, much attention has been paid to the bubbling problemfor semilinear Neumann elliptic equation with the critical and subcriticalpolynomial nonlinearity, much less is known if the polynomial nonlinearity isreplaced by the exponential nonlinearity. In this paper, we consider thefollowing semilinear Neumann elliptic problem with the Trudinger-Moserexponential growth: \begin{equation*}\begin{cases} -d\Deltau_d+u_d=u_d(e^{u^2_{d}}-1)\ \ \mbox{in}\ \Omega,\\ \frac{\partialu_d}{\partial\nu}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\\partial \Omega,\\ \end{cases}\end{equation*} where $d>0$ is a parameter,$\Omega$ is a smooth bounded domain in $\mathbb{R}^2$, $\nu$ is the unit outernormal to $\partial \Omega$. We first prove the existence of a ground statesolution to the above equation. If $d$ is sufficiently small, we prove that anyground state solution $u_d$ has at most one maximum point which is located onthe boundary of $\Omega$ and characterize the shape of ground state solution$u_d$ around the condensation point $P_d$. The key point of the proof lies inproving that the maximum point $P_d$ is close to the boundary at the speed of$\sqrt{d}$ when $d\rightarrow0$ and $u_d$ under suitable scaling transformconverges strongly to the ground state solution of the limit equation $\Deltaw+w=w(e^{w^2}-1)$.  Our proof is based on the energy threshold of cut-off function, theconcentration compactness principle for the  Trudinger-Moser inequality, regularity theory for elliptic equation and anaccurate analysis for the energy of the ground state solution $u_d$ as$d\rightarrow0$. Furthermore, by assuming that $\Omega$ is a unit disk, weremove the smallness assumption on $d$ and show the maximum point of groundstate solution $u_d$ must lie on the boundary of $\Omega$ for any $d>0$.

Bubbling phenomenon for semilinear Neumann elliptic equations of critical exponential growth