The Existence of Solutions to Elliptic Equations with Multiple Critical Nonlinearities and Rellich Potentials

In this paper, we investigate the Biharmonic problem that involves numerous critical non-linearities and Rellich potentials. \(\Delta^2 u-\mu_1 \frac{u}{|x|^4}-\mu_2 \frac{u}{|x-a|^4}=|u|^{2^*-2} u+\frac{|u|^{2^*(s)-2} u}{|x-a|^s} \quad \text { in } \Omega \backslash\{0, a\},\) where \(\Omega\) is a smooth open bounded domain in \(\mathbb{R}^N(N \geq 5)\), \(2^*(s)\)=\(\frac{2(N-s)}{N-4}\), 0 <s <4 and \(\mu_i<\bar{\mu}=\left[\frac{N(N-4)}{4}\right]^2(i=1,2)\). We use Mountain-Pass theorem to prove the existence of a solution to the above problem.