positive solutions to a semilinear elliptic equation with a sobolev–hardy term

elliptic equation with a sobolev–hardy term

{ −δu − λu =u^2♯−1/|y| x ∈ ω,

u > 0, x ∈ ω,

u ∈ h¹₀ (ω),

(0.1)

where ω is a bounded domain with smooth boundary in rⁿ (n ≥ 3), x = (y, z) ∈ ω ⊂

r^k × r^n−k = rⁿ , 2 ≤ k < n, 2^♯ := 2(n−1)/n−2 is the corresponding critical exponent and 0 < λ < λ₁ where λ₁ is the first eigenvalue of −δ in h¹₀ (ω). when n ≥ 4, we prove that problem (0.1) has at least one positive solution by using the mountain-pass lemma and a global compactness result. the case n = 3 is quite different and we deal with this case by using the method in jannelli (1999) [20] to prove the existence result. moreover, we obtain the nonexistence result of (0.1) in a star shaped domain. our main results extend a recent result of castorina et al. (2009) [10] where λ = 0 and ω = rⁿ .