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Nonlocal Lazer–McKenna-type problem perturbed by the Hardy’s potential and its parabolic equivalence

Bayrami Aminlouee, M ; Sharif University of Technology | 2021

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  1. Type of Document: Article
  2. DOI: 10.1186/s13661-021-01545-2
  3. Publisher: Springer Science and Business Media Deutschland GmbH , 2021
  4. Abstract:
  5. In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: {(−Δ)su=λu|x|2s+μuγ+fin Ω,u>0in Ω,u=0in (RN∖Ω). Here 0 < s< 1 , λ> 0 , γ> 0 , and Ω ⊂ RN (N> 2 s) is a bounded smooth domain such that 0 ∈ Ω. Moreover, 0 ≤ μ, f∈ L1(Ω). For 0 < λ≤ Λ N,s, Λ N,s being the best constant in the fractional Hardy inequality, we find a necessary and sufficient condition for the existence of a positive weak solution to the problem with respect to the data μ and f. Also, for a regular datum of f, under suitable assumptions, we obtain some existence and uniqueness results and calculate the rate of growth of solutions. Moreover, we mention a nonexistence and a complete blowup result for the case λ> Λ N,s. Besides, we consider the parabolic equivalence of the above problem in the case μ≡ 1 and some suitable f(x, t) , that is, {ut+(−Δ)su=λu|x|2s+1uγ+f(x,t)in Ω×(0,T),u>0in Ω×(0,T),u=0in (RN∖Ω)×(0,T),u(x,0)=u0in RN, where u0∈X0s(Ω) satisfies an appropriate cone condition. In the case 0 < γ≤ 1 or γ> 1 with 2 s(γ− 1) < (γ+ 1) , we show the existence of a unique solution for any 0 < λ< Λ N,s and prove a stabilization result for certain range of λ. © 2021, The Author(s)
  6. Keywords:
  7. Blowup ; Existence and nonexistence ; Hardy singularity ; Positive solution ; Singular fractional Laplacian heat equation ; Singular nonlinearity
  8. Source: Boundary Value Problems ; Volume 2021, Issue 1 , 2021 ; 16872762 (ISSN)
  9. URL: https://boundaryvalueproblems.springeropen.com/articles/10.1186/s13661-021-01545-2