2017/11/23 by Shaya Shakerian, Shakerian, Shaya
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1711.08839
arxiv created 2017/11/23 · arxiv updated 2017/11/27
Let Ω⊂ ℝn be a smooth bounded domain having zero in its interior 0 ∈ Ω. We fix 0 < α≤ 2 and 0 ≤ s <α. We investigate a sufficient condition for the existence of a positive solution for the following perturbed problem associated with the Hardy-Schrödinger operator Lγ,α,: = (- Δ)^\fracα2- \fracγ|x|α on Ω: \ (- Δ)^\fracα2u- γ(u)/(|x|α) - λu= \fracu2α^*(s)-1|x|s+ h(x) uq-1 · in Ω
u=0 · in ℝn ∖ Ω,. where 2α^*(s):=(2(n-s))/(n-α), λ∈ ℝ , h ∈ C0(Ω), h ≥ 0, q ∈ (2, 2^*α) with 2^*α:=2^*α(0), and γ< γH(α), the latter being the best constant in the Hardy inequality on ℝn. We prove that there exists a threshold γcrit(α) in ( - ∞, γH(α)) such that the existence of solutions of the above problem is guaranteed by the non-linear perturbation (i.e., h(x) uq-1) whenever γ≤ γcrit(α), while for γcrit(α)<γ<γH(α), it is determined by a subtle combination of the geometry of the domain and the size of the nonlinearity of the perturbations.