Existence of Solutions of a Non-Linear Eigenvalue Problem with a Variable Weight

2017/10/16 by Hadiji, Rejeb, Vigneron, Francois
#35A01 #35A15 #35J57 #35J62 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.05653

Abstract

We study the non-linear minimization problem on H¹₀(Ω)⊂ L^q with q=(2n)/(n-2), α>0 and n≥4~: inf_\substacku∈ H¹₀(Ω) ‖u‖_L^q=1∫_Ωa(x,u)|∇ u|² - λ∫_Ω |u|². where a(x,s) presents a global minimum α at (x₀,0) with x₀∈Ω. In order to describe the concentration of u(x) around x₀, one needs to calibrate the behaviour of a(x,s) with respect to s. The model case is inf_\substacku∈ H¹₀(Ω) ‖u‖_L^q=1∫_Ω(α+|x|^β|u|^k)|∇ u|² - λ∫_Ω |u|². In a previous paper dedicated to the same problem with λ=0, we showed that minimizers exist only in the range β kn/q + 2, minimizers do exist.

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