A Hopf's lemma and a strong minimum principle for the fractional p -Laplacian

Abstract

Our propose here is to provide a Hopf lemma and a strong minimum principle for weak supersolutions ofwhere Ω is an open set of R N , s ∈ (0, 1), p ∈ (1, +∞), c ∈ C(Ω) and (−∆ p ) s is the fractional p-Laplacian.

Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional <i>p</i>-Laplacian

“…which implies that u − = 0, that is u ≥ 0. Arguing as in Lemma 3.15 below, we can see that u ∈ L ∞ (R N ) ∩ C 0 (R N ), and by applying the maximum principle [16] we deduce that u > 0 in R N .…”

“…In view of Lemma 4. 16 we can see that v n → 0 in W ε , that is u n → u in W ε . Let us consider the case V ∞ = ∞.…”

On the multiplicity and concentration of positive solutions for a p-fractional Choquard equation in RN

“…Moreover, arguing as in Remark 3 in [10], we can prove that (−∆) s p ψ n + V 0 2 ψ p−1 n ≤ 0 holds for |x| ≥ R 2 . Thus, by the maximum principle [15], we infer that ψ n ≤ 0 for |x| ≥ R 2 , that is v n ≤ C 1 w for |x| ≥ R 2 . Recalling that w n (x) = u n ( x εn ) = v n ( x εn −ỹ n ) we can deduce that w n (x) = u n x ε n = v n x ε n −ỹ n…”

“…Arguing as in Lemma 6.5, we can obtain that v ∈ L ∞ (R N ) and applying Corollary 5.5 in [25] we have v ∈ C 0,α (R N ). Using the maximum principle in [15] we can conclude that u > 0 in R N .…”

Sobolev versus Hölder minimizers for the degenerate fractional p-Laplacian

“…Mainly, such choice is dictated by the results of [18] (see also [17,18]), where an a priori bound for solutions of (1.3) in the space C α s (Ω) = {u ∈ C α (Ω) : u/d s Ω ∈ C α (Ω)} for some α > 0, which compactly embeds into C 0 s (Ω) (see Section 2 for details). Plus, in [7] (see also [21]) the following version of Hopf's lemma was proved: any solution u 0 of (1.3) with non-negative right hand side either vanishes identically, or u/d s Ω c for some c > 0. In particular, these signed solutions belongs to the interior of the non-negative cone in C 0 s (Ω).…”