Uniqueness of positive solutions with critical exponent and inverse square potential (Q397191)

The paper is concerned with two critical exponent problems with Hardy potential (one almost critical and the other of perturbed type) NEWLINE\[NEWLINE\begin{cases}-\Delta u-\displaystyle\frac{\mu}{|x|^2}u=u^{2^*-1-\varepsilon}\quad & \text{in }\Omega,\\ u>0\quad & \text{in }\Omega,\\ u=0\quad & \text{on }\partial \Omega,\end{cases}\leqno(1)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\begin{cases}-\Delta u-\displaystyle\frac{\mu}{|x|^2}u=u^{2^*-1}+\varepsilon u\quad & \text{in }\Omega,\\ u>0\quad & \text{in }\Omega,\\ u=0\quad & \text{on }\partial \Omega,\end{cases}\leqno(2)NEWLINE\]NEWLINE where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) \((N\geq 3)\), \(0\in \Omega\), \(0<\mu<\frac{(N-2)^2}{4}\), and \(\varepsilon>0\) is a parameter. Under some additional assumptions, the author proves the local uniqueness of positive solutions for \((1)\) and \((2)\), by analyzing the associated Green function and the blow up profile of their positive solutions. The critical constant of the blow up behavior of solutions is also determined.