A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in RN

Abstract In this paper, we will study the following PDE in R N involving multiple Hardy-Sobolev critical exponents: { Δ u + ∑ i = 1 l λ i u 2 ⁎ ( s i ) − 1 | x | s i + u 2 ⁎ − 1 = 0 in R N , u ∈ D 0 1 , 2 ( R N ) , where 0 s 1 s 2 ⋯ s l 2 , 2 ⁎ : = 2 N N − 2 , 2 ⁎ ( s ) : = 2 ( N − s ) N − 2 and there exists some k ∈ { 1 , ⋯ , l } such that λ i > 0 for 1 ≤ i ≤ k ; λ i 0 for k + 1 ≤ i ≤ l . We develop an interesting way to study this class of equations involving mixed sign parameters. We prove the existence of the positive ground state solution. The regularity of the least-energy solution is also investigated.