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

Abstract

The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents −Δu−μux2=u2*s−2xsu+g(x,u),x∈Ω∖0,u=0,x∈∂Ω have been investigated, where Ω is an open-bounded domain in RNN≥3, with a smooth boundary ∂Ω, 0∈Ω,0≤μ<μ¯:=N−222,0≤s<2, and 2*s=2N−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 PSc conditions, we obtained that: (a) If μ≤μ¯−1, and λ<λ1μ, then the above problem has at least a positive solution in H01Ω; (b) If μ¯−1<μ<μ¯, then when λ*μ<λ<λ1μ, the above problem has at least a positive solution in H01Ω; (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)=λu).