A non-local semilinear eigenvalue problem

Abstract

We prove that positive solutions of the fractional Lane–Emden equation with homogeneous Dirichlet boundary conditions satisfy pointwise estimates in terms of the best constant in Poincaré’s inequality on all open sets, and are isolated in $$L^1$$ L 1 on smooth bounded ones, whence we deduce the isolation of the first non-local semilinear eigenvalue.

Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

“…Consequently, the global minimizer û is nonzero in Ω. In view of the evenness of E, û can be assumed nonnegative, and hence the strong maximum principle (see, e.g., [15,Proposition A.2]) guarantees that û > 0 a.e. in Ω.…”

“…in Ω. Note that [15, Proposition A.2] is applicable since the space D s,2 0 (Ω) coincides with X s 0 (Ω) under our assumptions on Ω, see [15,Remark 2.1]. In general, we refer to [15] for a thorough discussion on the fractional Lane-Emden problem (3.18) in the unweighted case q = 1 a.e in Ω.…”

“…Note that [15, Proposition A.2] is applicable since the space D s,2 0 (Ω) coincides with X s 0 (Ω) under our assumptions on Ω, see [15,Remark 2.1]. In general, we refer to [15] for a thorough discussion on the fractional Lane-Emden problem (3.18) in the unweighted case q = 1 a.e in Ω. Moreover, we have û ∈ L ∞ (Ω), see, e.g., [ The positive global minimizer û of E generates a stationary bounded finite energy solution (and hence supersolution) of (1.1).…”

Large time behavior of fractional porous media equation

“…Given a solution u of (1.1), the equation (1.5) for the function v(x, t) = e αt u(x, e t − 1) describes a system that evolves, irrespective of the starting conditions, to fixed points, i.e., states of the form Φ −1 (v) with v being a critical point of the energy functional. Because of the isolation of the energy minimizing solutions ±w Ω , proved in [16], this and the Lyapunov property imply that the disconnected set {±Φ −1 (w Ω )} has a non-empty basin of attraction, including all initial states with energy smaller than the first excited level. (The isolation property implies some restriction on boundary regularity: assumptions weaker than those made in the our main statement are also feasible, but that is not the object of this paper.…”

“…with homogeneous Dirichlet boundary conditions. It is known [16] that the minimal energy (1.4) Λ 1 = min{F s q,α (ϕ) : ϕ ∈ D s,2 0 (Ω)} , is achieved by a solution with constant sign, that it is unique (up to the sign). Also, we set Φ(u) = |u| m−1 u and we observe that Φ −1 (ϕ) = |ϕ| q−2 ϕ where q = (m + 1)/m.…”