Existence and multiplicity results for resonant fractional boundary value problems

Abstract

We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.2010 Mathematics Subject Classification. 35R11, 35P05, 49F15.

Analysis of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with p‐Laplacian in Banach space

“… is the Caputo fractional derivative while is the Riemann‐Liouville fractional derivative. For some more related and useful works, we suggest the readers to Liu and Chen, Caballero et al, Lakoud and Ashyralyev, and Iannizzotto and Papageorgiou …”

Nontrivial solutions to non-local problems with sublinear or superlinear nonlinearities

“…We notice that since aðxÞ 2 L N 2s ðXÞ is not necessarily bounded, eigenvalue problem (7) does not seem to have been investigated in the literature. Hence we will first study problem (7). Particularly, we will prove that it has and only has a sequence of eigenvalues k 1 \k 2 \k 3 \ Á Á Á \k n \::: with a finite multiplicity for each eigenvalue.…”

“…When g(x, u) is a lower order perturbation of the critical power, the classical Brezis-Nirenberg results are established in [5]. Some multiplicity results are also established either by Morse theory eg see [6,7] or by fountain theorems eg see [8] where superlinear nonlinearities without Ambrosetti-Rabinowitz conditions are considered.…”

Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

“…The problem (1.1) with ρ ≡ 1 has been investigated by Servadei and Valdinoci in [27] for a general nonlocal operator. Molica Bisci et al, in [24], studied the same problem with a positive and Lipschitz continuous weight ρ. Iannizzotto and Papageorgiou, in [21], considered the case of a general positive function ρ ∈ L ∞ (Ω) and Frassu and Iannizzotto, in [18], treated a more general eigenvalue problem with indefinite weight ρ ∈ L ∞ (Ω). We denote by λ k (ρ), k ∈ Z {0}, the k-th eigenvalue of problem (1.1) corresponding to the weight ρ.…”