Bubbling solutions for an anisotropic Emden–Fowler equation

Abstract

We consider the following anisotropic Emden–Fowler equation \(\nabla (a(x) \nabla u)+ \epsilon^{2} a(x) e^{u} = 0 \quad in \quad \Omega, \quad u=0 \quad on \quad \partial \Omega,\) where \(\Omega \subset \mathbb{R}^2\) is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that at given local maximum points of a(x), there exists arbitrarily many bubbles. As a consequence, the quantity \(\mathcal{T}_\epsilon = \epsilon^{2} \int_{\Omega} a(x)e^{u} {\rm d}x \) can approach to \( + \infty\) as \(\epsilon \to 0\). These results show a striking difference with the isotropic case [\(a(x) \equiv \) Constant].