Singular Nonlocal Problem Involving Measure Data

Singular Fractional Choquard Equation with a Critical Nonlinearity and a Radon measure

“…According to Lemma 2.3 and Lemma 2.4 of Ghosh et.al [17], (3.15) admits a nontrivial positive weak solution in H s 0 (Ω) and for any ω ⊂⊂ Ω, there exists C ω such that w n ≥ C ω > 0. Using a standard comparison principle, Lemma 2.4 of [1], we conclude that w n ≤ w n .…”

“…The local case (with Laplace operator) of such problems has been dealt by Panda et al in [27] and the corresponding problem admits a weak solution in W 1,m 0 (Ω) if γ ∈ (0, 1] and in W 1,m loc (Ω) if γ > 1 for all m < N N −1 . The nonlocal case (with fractional Laplacian) with a singularity and a Radon measure has been studied by Ghosh et al in [17]. In this paper we will consider the following singular fractional elliptic problem with a Choquard type critical nonlinearity and a Radon measure.…”

Nonlocal Lazer-McKenna type problem perturbed by the Hardy's potential and its parabolic equivalence

“…where Ω is a bounded domain of R N , N ≥ 2, γ > 0 and µ a general Radon measure in Ω. Also, see the papers [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41] for more related problems. These types of problems have been extensively studied for their relations with some physical phenomena in the theory of pseudoplastic fluids, [42].…”

A fractional Laplacian problem with mixed singular nonlinearities and nonregular data

“…In fact, in the case 0 < γ ≤ 1, and µ ∈ L 1 (Ω), we will show the existence and uniqueness of the entropy solution to problem (1) for some weighted integrable functions f . It is important to note that in [8,Sections 3 and 4], the authors discussed uniqueness for the notion of very weak solution to problem (1) by invoking a Kato type inequality in the case 0 < γ < 1, with K ≡ 0. Also, see the article [9], where the authors investigated the existence and uniqueness of a notion of a solution for the fractional p-Laplacian case of (1) with K, µ ≡ 0.…”