Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents in RNRN

Let N≥3,0≤s<2,0≤μ<(N−22)2 and 2∗(s)≔2(N−s)N−2 be the critical Sobolev–Hardy exponents. Via variational methods and the analytic technique, we prove the existence of a nontrivial solution to the singular semilinear problem −Δu−μu|x|2+u=|u|2∗(s)−2|x|su+f(u),u∈Hr1(RN), for N≥4,0≤μ≤μ̄−1 and suitable functions f(u)f(u).