Existence of solutions of a non-linear eigenvalue problem with a variable weight

Abstract

We study the non-linear minimization problem on H01(Ω)⊂Lq with q=2nn−2, α>0 and n≥4:infu∈H01(Ω)‖u‖Lq=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 behavior of a(x,s) with respect to s. The model case isinfu∈H01(Ω)‖u‖Lq=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, which corresponds to a dominant non-linear term. On the contrary, the linear influence for β≥kn/q prevented their existence. The goal of this present paper is to show that for 0<λ≤αλ1(Ω), 0≤k≤q−2 and β>kn/q+2, minimizers do exist.