Boundary concentration phenomena for an anisotropic Neumann problem in ${\mathbb{R}}^2$

Abstract

Given a smooth bounded domain $\Omega$ in ${\mathbb{R}}^2$, we study the following anisotropic Neumann problem $$\{\begin{array}{l}-\nabla (a(x)\nabla u)+a(x)u=\lambda a(x)u^{p-1}{e}^{u^p},u>0\text{in}\Omega ,\\ \frac{\partial u}{\partial \nu }=0\text{on}\partial \Omega ,\end{array}$$ where $\lambda >0$ is a small parameter, $0<p<2$, $a(x)$ is a positive smooth function over $\overline{\Omega }$ and $\nu$ denotes the outer unit normal vector to $\partial \Omega$. Under suitable assumptions on anisotropic coefficient $a(x)$, we construct solutions $u_{\lambda }$ 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 $\partial \Omega$, or accumulate to the same strict local maximum boundary point of $a(x)$ over $\overline{\Omega }$ as $\lambda \to 0$. Furthermore, for these bubbling solutions $u_{\lambda }$ we compute the delicate expansion of the corresponding energy $${\beta }_{\lambda }=\frac{\lambda p}{2}{\left({\int }_{\Omega }a(x)(e^{u_{\lambda }^p}-1)dx\right)}^{\frac{2-p}{p}}{\left({\int }_{\Omega }a(x)u_{\lambda }^p{e}^{u_{\lambda }^p}dx\right)}^{\frac{2(p-1)}{p}}$$ and exhibit the sharp difference of tendency of energy ${\beta }_{\lambda }$ between $0<p<1$ and $1<p<2$.