Boundary concentration phenomena for an anisotropic Neumann problem in $\mathbb{R}^2$
Given a smooth bounded domain $Ω$ in $\mathbb{R}^2$, we study the following anisotropic Neumann problem $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=λa(x) u^{p-1}e^{u^p},\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, Ω,\\[2mm] \frac{\partial u}{\partialν}=0\,\, \qquad\quad\qquad\qquad\qquad\qquad\qquad \ \ \ \ \,\qquad\quad\, \textrm{on}\,\,\, \partialΩ, \end{cases} $$ where $λ>0$ is a small parameter, $0<p<2$, $a(x)$ is a positive smooth function over $\overlineΩ$ and $ν$ denotes the outer unit normal vector to $\partialΩ$. Under suitable assumptions on anisotropic coefficient $a(x)$, we construct solutions $u_λ$ of this problem with arbitrarily many mixed interior and boundary bubbles which concentrate at totally different strict local maximum or minimal boundary points of $a(x)$ restricted to $\partialΩ$, or accumulate to the same strict local maximum boundary point of $a(x)$ over $\overlineΩ$ as $λ\rightarrow0$. Furthermore, for these bubbling solutions $u_λ$ we compute the delicate expansion of the corresponding energy $$ β_λ=\,\frac{λp}{2} \left( \int_Ω a(x)\big(e^{u_λ^p}-1\big)dx \right)^{\frac{2-p}{p}} \left( \int_Ω a(x)u_λ^p e^{u_λ^p}dx \right)^{\frac{2(p-1)}{p}} $$ and exhibit the sharp difference of tendency of energy $β_λ$ between $0<p<1$ and $1<p<2$.