In this paper, we investigate the existence of multiple positive solutions to following multi-critical elliptic problem $$\begin{aligned} \left\{ \begin{aligned} -\Delta u&=\lambda |u|^{p-2}u +\sum _{i=1}^k(|x|^{-(N-\alpha _i)}*|u|^{2^*_i})|u|^{2^*_i-2}u\quad \mathrm{in}\quad \Omega ,\\&u\in H^1_0(\Omega )\\ \end{aligned}\right. \end{aligned}$$ (0.1) in connection with topology bounded domain \...

Consider a large linear system where $A_n$ is $n\times n$ matrix with independent real standard Gaussian entries, $\boldsymbol{1}_n$ 1$ vector of ones and unknown the $\boldsymbol{x}_n$ satisfying $$ \boldsymbol{x}_n = \boldsymbol{1}_n +\frac 1{\alpha_n\sqrt{n}} A_n \boldsymbol{x}_n\, . $$ We investigate (componentwise) positivity solution depending on scaling factor $\alpha_n$ as dimension...