Bubbling solutions to an anisotropic Hénon equation

Abstract

In this paper we consider the problem

$$\displaystyle{ \qquad \left \{\begin{array}{ll} -\mathrm{div}(a(x)\nabla u) = c_{\alpha }a(x)\vert x -\xi \vert ^{\alpha }u^{p_{\alpha }\pm \varepsilon }&\mbox{ in }\varOmega, \\ u > 0 &\mbox{ in }\varOmega, \\ u = 0 &\mbox{ on }\partial \varOmega,\end{array} \right. }$$

(1)

where Ω is a bounded smooth domain in $\mathbb{R}^{N}$, N ≥ 3, the function $a \in C^{2}(\overline{\varOmega })$ is strictly positive on $\overline{\varOmega }$, $\xi \in \varOmega$, $p_{\alpha }:= \frac{N+2+2\alpha } {N-2}$, $\varepsilon > 0$, c _α : = (N +α)(N − 2) and α is a positive real number which is not an even integer. Here $\varepsilon$ is a positive small parameter. We give some sufficient conditions on the function a which ensure existence of solutions to (1) blowing up at the point $\xi$ as $\varepsilon$ goes to zero.