Singular limits in higher order Lioville-type equations

الأصل · EN

In this paper we consider the higher order Lioville-type equation (-Δ)ᵐ u=ρ²ᵐ V(x) eᵘ in Ω²ᵐ with V≠0 a given smooth potential, ρ⁺ a small parameter which tends to zero from above and where we prescribe the boundary conditions to be either Navier or Dirichlet. We find sufficient conditions under which, as ρ approaches 0, there exists an explicit class of solutions which admit a concentration behavior with a prescribed bubble profile around some given k-points in Ω, for any given integer k. These are the so-called singular limits. The candidate k-points of concentration must be critical points of a suitable finite dimensional functional explicitly defined in terms of the potential V and the higher order Green's function with respect to the imposed boundary conditions.