Necessary condition in a Brezis–Oswald-type problem for mixed local and nonlocal operators

Optimal solvability for the fractional p-Laplacian with Dirichlet conditions

“…1 , as claimed. Moreover, by letting t → 0 + in (2.3) and recalling the extended definition of f , we have for a.e.x ∈ Ω and all t ∈ R −…”

“…(a) The boundedness condition H (i) is obviously satisfied in the autonomous case, i.e., f ∈ C(R + ). (b) The growth condition H (ii) acts on the reaction from above only, as is the case in [6,9] but differently from previous results in the nonlocal setting such as [1,2,25]. (c) The strict monotonicity condition H (iii) classifies our reaction as a (p − 1)-sublinear one.…”

“…In the nonlocal framework, we recall the related results for the fractional Laplacian in R N [27], for the spectral fractional Laplacian in a bounded domain [23], for the fractional p-Laplacian with nonlocal Robin conditions [25], and a mixed local-nonlocal operator with Dirichlet conditions [1,2]. We remark again that the last three contributions present partial results regarding the necessary condition for existence, and they both employ bilateral growth conditions on the reaction.…”

“…This is a nonlinear, nonlocal operator which in special case reduces, up to a multiplicative constant, to the fractional Laplacian (p = 2, s ∈ (0, 1)), to the p-Laplacian (p > 1, s = 1), and in particular to the Laplacian (p = 2, s = 1). The fractional p-Laplacian is degenerate if p > 2, singular if p ∈ (1,2). An introduction to this operator and related problems can be found in [24,26].…”

Variational methods for nonpositive mixed local-nonlocal operators

“…[2,9,13,17,24,25]), boundary Harnack principle [14], boundary regularity and overdetermined problems [10,36], qualitative properties of solutions [3], existence of solutions and asymptotics (see e.g. [6,7,8,12,15,20,21,26,33,35]) and shape optimization problems [4,5,27].…”