Four solutions of an inhomogeneous elliptic equation with critical exponent and singular term

Abstract

In this paper, we prove the existence of four nontrivial solutions of − Δ u − λ | x | 2 u = | u | 2 ∗ − 2 u + μ | x | α − 2 u + f ( x ) , x ∈ Ω ∖ { 0 } and show that at least one of them is sign changing. Our results extend some previous works, such as [G. Tarantello, Multiplicity results for an inhomogeneous Neumann problem with critical exponent, Manuscripta Math., 81 (1993) 51–78; D. Kang and Y. Deng, Multiple solutions for inhomogeneous elliptic problems involving critical Sobolev–Hardy exponents, J. Math. Anal. Appl., 60 (2005) 729–753; N. Hirano and N. Shioji, A multiplicty result including a sign changing solution for an inhomogeneous Neumann problem with critical exponent, Proc. Roy. Soc. Edinburgh, 137A (2007) 333–347.]