Abstract
In this paper, we study the existence and nonexistence of solutions to fractional elliptic equations with the Hardy potential −Δsu−λu/x2s=ur−1+δgu,in Ω,ux>0,in Ω,ux=0,in ℝN∖Ω, where Ω⊂ℝN is a bounded Lipschitz domain with 0∈Ω, −Δs is a fractional Laplace operator, s∈0,1, N>2s, δ is a positive number, 2<r<rλ,s≡N+2s−2αλ/N−2s−2αλ+1, αλ∈0,N−2s/2 is a parameter depending on λ, 0<λ<ΛN,s, and ΛN,s=22sΓ2N+2s/4/Γ2N−2s/4 is the sharp constant of the Hardy–Sobolev inequality.
Highlights
In this paper, we consider the solvability of the following fractional elliptic problem:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨ (− Δ)su − λ u |x|2s ur− + δg(u), in Ω, ⎪⎪⎪⎪⎪⎪⎪⎪⎩ u u > 0, 0, in Ω, (1)in RN \ Ω, where Ω ⊂ RN is a bounded Lipschitz domain with 0 ∈ Ω, s ∈ (0, 1), N > 2s, 2
We study the existence and nonexistence of solutions to fractional elliptic equations with the Hardy potential
We consider the solvability of the following fractional elliptic problem:
Summary
We consider the solvability of the following fractional elliptic problem:. Shang et al [10] studied the existence and multiplicity of positive solutions to the following problem:. En, there exists a positive constant δ0, such that, for all δ ∈ (λ1/M0, δ0], problem (1) has at least a nonnegative solution if M0 > (λ1/δ0), where M0 is defined by (12). An example of function g(σ) ≡ σq with 0 < q < 1, which satisfies conditions (10) and (11) for any δ ∈ (0, δ0], such that problem (1) has at least one positive solution. In this condition ηr− qc−11C21− q(C1 −.
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