A minimizing problem of a polyharmonic operator with Critical Exponent

2022/02/18 by Hadiji, Asma Benhamida Rejeb, Yazidi, Habib
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.09404

Abstract

In this work, we study the two following minimization problems for r ∈ ℕ^*, S_0,r(φ)=inf_u∈ H₀^r(Ω) |u+φ‖_L^2^*r=1‖u‖_r² · amp; \textrmand · amp; S_θ,r(φ)=inf_u∈ H_θ^r(Ω) ‖u+φ‖_L^2^*r=1‖u‖_r², where Ω⊂ ℝ^N, N > 2r, is a smooth bounded domain, 2^*r=(2N)/(N-2 r), φ∈ L^2^*r (Ω) ∩ C(Ω) and the norm ‖. ‖_r= ∫_Ω |(-Δ)^α .|²dx where α=(r)/(2) if r is even and ‖. ‖_r= ∫_Ω |∇(-Δ)^α . |²dx where α= (r-1)/(2) if r is odd. Firstly, we prove that, when φ\not≡ 0, the infimum in S_0,r(φ) and S_θ,r(φ) are attained. Secondly, we show that S_θ,r(φ)< S_0,r(φ) for a large class of φ.

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