Journal Articles Advances in Calculus of Variation Year : 2018

Stationary Kirchhoff equations with powers

Emmanuel Hebey
  • Function : Author

Abstract

Abstract We discuss existence of solutions, compactness and stability properties in closed manifolds for the critical Kirchhoff equations ( a + b ∫ M | ∇ u | 2 d v g ) θ 0 Δ g u + h u = u p - 1 , \Bigg{(}a+b\int_{M}\lvert\nabla u|^{2}\,dv_{g}\Bigg{)}^{\theta_{0}}\Delta_{g}u% +hu=u^{p-1}, where Δ g {\Delta_{g}} is the Laplace–Beltrami operator, h is a C 1 {C^{1}} -function in M , p ∈ ( 2 , 2 ⋆ ] {p\in(2,2^{\star}]} , a , b , θ 0 > 0 {a,b,\theta_{0}>0} are positive real numbers, and 2 ⋆ {2^{\star}} is the critical Sobolev exponent. A fractional critical dimension d 0 = 2 ⁢ ( 1 + θ 0 ) θ 0 {d_{0}=\frac{2(1+\theta_{0})}{\theta_{0}}} appears in the critical case p = 2 ⋆ {p=2^{\star}} .
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Dates and versions

hal-04817420 , version 1 (03-12-2024)

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Emmanuel Hebey. Stationary Kirchhoff equations with powers. Advances in Calculus of Variation, 2018, 11 (2), pp.139-160. ⟨10.1515/acv-2016-0025⟩. ⟨hal-04817420⟩

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