Singular fourth-order elliptic equations on~$\smash{\mathbb{R}^N}$: Existence and multiplicity of solutions

Abstract This paper establishes some existence results for a class of singular fourth-order elliptic equations on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {\mathbb{R}^{N}} , featuring a degenerate potential <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>δ</m:mi> </m:mrow> </m:mrow> </m:msup> </m:math> {|x|^{-2\delta}} and a singular nonlinearity. The profound lack of compactness poses a significant challenge. To overcome this, we develop a perturbative approach. Our first main result proves the existence of infinitely many solutions in the case of an odd nonlinearity. Finally, a careful limiting argument connects these results from the perturbed problem back to the original singular equation.