Remarks on solutions of a fourth-order problem

In this paper, we study the two following minimization problems: S0(q,φ)=infu∈H02(Ω),‖u+φ‖q=1∫Ω|Δu|2andSθ(q,φ)=infu∈Hθ2(Ω),‖u+φ‖q=1∫Ω|Δu|2. We prove that for a class of maps φφ, we have Sθ(q,φ)