Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents − Δ u − μ u x 2 = u 2 * s − 2 x s u + g ( x , u ) , x ∈ Ω ∖ 0 , u = 0 , x ∈ ∂ Ω have been investigated, where Ω is an open-bounded domain in R N N ≥ 3 , with a smooth boundary ∂ Ω , 0 ∈ Ω , 0 ≤ μ < μ ¯ : = N − 2 2 2 , 0 ≤ s < 2 , and 2 * s = 2 N − s / N − 2 is the Hardy–Sobolev critical exponent. This problem comes from the study of standing waves in the anisotropic Schrödinger equation; it is very important in the fields of hydrodynamics, glaciology, quantum field theory, and statistical mechanics. Under some deterministic conditions on g , by a detailed estimation of the extremum function and using mountain pass lemma with P S c conditions, we obtained that: (a) If μ ≤ μ ¯ − 1 , and λ < λ 1 μ , then the above problem has at least a positive solution in H 0 1 Ω ; (b) If μ ¯ − 1 < μ < μ ¯ , then when λ * μ < λ < λ 1 μ , the above problem has at least a positive solution in H 0 1 Ω ; (c) if μ ¯ − 1 < μ < μ ¯ and Ω = B ( 0 , R ) , then the above problem has no positive solution for λ ≤ λ * μ . These results are extensions of E. Jannelli’s research ( g ( x , u ) = λ u ).