Bubbling solutions to an anisotropic Hénon equation

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.