The effect of topology on the number of positive solutions of elliptic equation involving Hardy-Littlewood-Sobolev critical exponent (Q2178782)

The following elliptic Choquard type equation is considered \[ -\Delta u=\lambda |u|^{q-2}u+\left( \int_\Omega \frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}\,dy\right)|u|^{2^*_\mu -2 }u, \quad u>0 \text{ in } \Omega, \] with \(u=0\) on \(\partial \Omega\), where \(\Omega\) ia an open bounded set with continuous boundary in \(\mathbb R^N\) \(( N\geq 3)\), \(2^*_\mu=(2N-\mu)/N-2\) and \(q\in [2,2N/(N-2))\). The main result states that for sufficiently small \(\lambda\) the problem has at least \(Cat (\Omega)\) positive solutions under one of the following conditions: \begin{itemize} \item[(a)] \(N>3\) and \(q\in [2,2N/(N-2))\), or \item[(b)] \(N=3\) and \(4<q<6\). \end{itemize} \(\mathrm{Cat}(\Omega)\) is the Ljusternik-Schnirelman category: \[ \mathrm{Cat} (\Omega)=\min \{ k\in \mathbb N : \exists \text{ closed } Y_1,\dots,Y_k \subset \Omega \text{ contractible in } \Omega \text{ and } \cup_{i=1}^kY_i=\Omega \}. \]