Global Bifurcation for Fractional $p$-Laplacian and an Application

Abstract

Abstract. We prove the existence of an unbounded branch of solutions to the non-linear non-local equationbifurcating from the first eigenvalue. Here (−∆) s p denotes the fractional p-Laplacian and Ω ⊂ R n is a bounded regular domain. The proof of the bifurcation results relies in computing the Leray-Schauder degree by making an homotopy respect to s (the order of the fractional p-Laplacian) and then to use results of local case (that is s = 1) found in [17]. Finally, we give some application to an existence result.

Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian

“…We note that Proposition 2.1 is a special case (p = 2) of Lemma 4.6 in [13] and we omit the proof here.…”

“…In fact, any constant great than or equal to M = maxΩ[a(x)/b(x)] 1/(p−1) is a super-solution. Let φ be a positive eigenfunction corresponding to µ 1 (for the existence of the first eigenvalue and corresponding eigenfunction has been obtained in [13] and [15]), then for each fixed µ > µ 1 and small positive ε, εφ < M and is a sub-solution. Therefore, by the sub-and super-solution method (see [14]), there exist at least one positive solution.…”

Eigenvalues for A Combination Between Local and Nonlocal p-Laplacians

“…For references concerning nonlocal fractional problems we refer to [7,20,22,12] and references therein. For the eigenvalue problem for this operator we refer to [6,9,10,11,13,22,14] where a detailed study was carry over showing similarities and differences with the local case (one of the biggest differences is that the restriction of an eigenfunction to a nodal domain is not an eigenfunction of this nodal domain due to the nonlocal character of the problem). Also in [22] the limit case p → ∞ was studied in the fractional setting.…”

Properties of eigenvalues and some regularities on fractionalp-Laplacian with singular weights

“…Perera-Sim [23] introduced a class B q , for q ∈ [1, p * ) where p * := N p N −p if p < N and p * := ∞ if p ≥ N, the class of measurable functions K such that for the distance function ρ (see (2.2)), Kρ a ∈ L r (Ω) for some a ∈ [0, q − 1] and r ∈ (1, ∞) satisfying has been the center of PDEs since such problems arised in various fields [1,6]. The fractional p-Laplace eigenvalue problems have mostly been studied under at most an L ∞ -weight (see [5,9,15,16,20]). Recently, Ho-Perera-Sim-Squassina [19] if ps < N and p * := ∞ if p ≥ N) to B q in the p-Laplacian eigenvalue problem and obtained the existence of the first eigenpair (λ 1 , e 1 ).…”