2019
DOI: 10.1063/1.5107517
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Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity

Abstract: In this paper we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given aswhere Ω is an open bounded domain in R N with smooth boundary, N > ps, s ∈ (0, 1), λ > 0, 0 < γ < 1, 1 < p < ∞, p − 1 < q ≤ p * s = N p N −ps . We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some β ∈ (0, 1), the weak solution to the problem is in C 1,β (Ω).

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Cited by 33 publications
(19 citation statements)
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“…This proves that w α is a weak solution to (3.6). By the weak comparison principle for fractional p-Laplacian (Lemma 3.1, [22]), we conclude that u λ ≤ w α a.e. in Ω.…”
Section: Existence Results Formentioning
confidence: 82%
See 1 more Smart Citation
“…This proves that w α is a weak solution to (3.6). By the weak comparison principle for fractional p-Laplacian (Lemma 3.1, [22]), we conclude that u λ ≤ w α a.e. in Ω.…”
Section: Existence Results Formentioning
confidence: 82%
“…have been applied to study the problems of type (1.4) for both local and nonlocal cases. Refer [15,21,22,23,24,25,27,28,30,31] and the bibliography therein. The main result of this article is the following.…”
Section: Introductionmentioning
confidence: 99%
“…Thus, the problem discussed here is new as the consideration of a singularity with a critical exponent with µ ∈ R handled with Morse theoretic approach is not found anywhere in the literature to our knowledge. The question about the existence and multiplicity of positive weak solutions to problem (1.1) with µ > 0 and with either a = 0 or b = 0 has been answered in [17,18,19,22,23,31] and the references therein. The authors of these works followed different tools such as variational method, concentration compactness method, Nehari manifold method etc.…”
Section: ˆRnmentioning
confidence: 99%
“…Moreover, one can proceed on similar lines as in cf. [20,Lemma 6.4] to conclude that I λ is C 1 . Since the functional J λ is Gâteaux differentiable, then the set of critical points of I λ coincides with the same for J λ .…”
Section: Preliminaries and Weak Formulationsmentioning
confidence: 99%