Sobolev versus Hölder minimizers for the degenerate fractional p-Laplacian

Abstract

We consider a nonlinear pseudo-differential equation driven by the fractional p-Laplacian (−∆) s p with s ∈ (0, 1) and p 2 (degenerate case), under Dirichlet type conditions in a smooth domain Ω. We prove that local minimizers of the associated energy functional in the fractional Sobolev space W s,p 0 (Ω) and in the weighted Hölder space C 0 s (Ω), respectively, do coincide.2010 Mathematics Subject Classification. 35D10, 35R11, 47G20.

Four Solutions for Fractional p-Laplacian Equations with Asymmetric Reactions

“…Our result represents an application of classical methods in nonlinear analysis combined with the recently established theory for the fractional p-Laplacian (mainly the results of [10,15,22]). To our knowledge, this is the first multiplicity result for a fractional order problem with asymmetric reaction, even in the linear case p = 2.…”

“…In the quasilinear case p = 2, things are obviously more involved. The eigenvalue problem for (−Δ) s p was first studied in [25], variational methods for equations with several types of reactions were established in [18], Hölder regularity of weak solutions was studied in [20,21] (for p > 2), maximum and comparison principles were proved in [10,23], equivalence between Sobolev and Hölder minimizers of the energy functional was proved in [22], and a detailed study of sub-and supersolutions was performed in [15]. Existence results for the fractional p-Laplacian with asymmetric reactions were obtained in [17,35], while closely related problems were studied in [1][2][3]7,11,40].…”

Regularity results for Choquard equations involving fractional $p$-Laplacian

“…In the fractional framework (p = 2), the analogous result is proved in [21]. In [22], this result is further generalized for the fractional p-Laplacian set up for p ≥ 2. In case of nonlocal nonlinearity, in particular, Choquard equation, Sreenadh et.…”

Extremal constant sign solutions and nodal solutions for the fractional p-Laplacian

“…The nonlinear case is obviously more involved: spectral properties of (−∆) s p were studied in [4,17,18,21,30], a detailed regularity theory was developed in [3,24,25,28,29] (some results about Sobolev and Hölder regularity being only proved for the degenerate case p > 2), maximum and comparison principles have appeared in [12,27], while existence and multiplicity of solutions have been obtained for instance in [10,13,18,22,39] (see also the surveys [32,35]). For the purposes of the present study, we recall in particular [26], where it was proved that the local minimizers of the energy functional corresponding to problem (1.1) in the topologies of W s,p 0 (Ω) and of the weighted Hölder space C 0 s (Ω), respectively, coincide (namely, a nonlinear fractional analogue of the classical result of [5]).…”

“…Remark 5.2. The argument based on the characterization of λ 2 was already employed in [26,Theorem 4.1] and [15,Theorem 3.3] (for p = 2). The novelty of Theorem 5.1 above, with respect to such results (even for the linear case p = 2), lies in the detailed information about solutions, as we prove that u ± are extremal constant sign solutions and ũ is nodal.…”