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. \(Δ^2 u-μ_1 u/|x|^4-μ_2 u/|x-a|^4=|u|2^*-2 u+{|u|2^*(s)-2 u}{|x-a|^s} in Ω \{0, a\},\) where \(Ω\) is a smooth open bounded domain in \(R^N(N ≥ 5)\), \(2^*(s)\)=\(2(N-s)/N-4\), 0 <s <4 and \(μ_i<μ̄=[N(N-4)/4]^2(i=1,2)\). We use Mountain-Pass theorem to prove the existence of a solution to the above problem.