Existence of positive solutions for a fractional elliptic problems with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities

Cai, Zhipeng; Chu, Changmu; Lei, Chun‐Yu

doi:10.1186/s13661-017-0912-8

Cited by 2 publications

(1 citation statement)

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“…Nowadays, researchers are inspecting on various forms of singular nonlocal equations. We cite [10,7,8] as some contemporary woks related to it. Our paper has been organized as follows-Section 2 contains the function space setting along with some preliminary results.…”

Section: Introductionmentioning

confidence: 99%

A Global multiplicity result for a very singular critical nonlocal equation

Giacomoni¹,

Mukherjee²,

Sreenadh³

2019

TMNA

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In this article, we show the global multiplicity result for the following nonlocal singular problemwhere Ω is a bounded domain in R n with smooth boundary ∂Ω, n > 2s, s ∈ (0, 1), λ > 0, q > 0 satisfies q(2s − 1) < (2s + 1) and 2 * s = 2n n−2s . Employing the variational method, we show the existence of at least two distinct weak positive solutions for (P λ ) in X 0 when λ ∈ (0, Λ) and no solution when λ > Λ, where Λ > 0 is appropriately chosen. We also prove a result of independent interest that any weak solution to (P λ ) is in C α (R n ) with α = α(s, q) ∈ (0, 1). The asymptotic behaviour of weak solutions reveals that this result is sharp.

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Section: Introductionmentioning

confidence: 99%

A Global multiplicity result for a very singular critical nonlocal equation

Giacomoni¹,

Mukherjee²,

Sreenadh³

2019

TMNA

View full text Add to dashboard Cite

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Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

Shen

2019

Advanced Nonlinear Studies

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In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

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Existence of positive solutions for a fractional elliptic problems with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities

Abstract: In this paper, we consider the study of a fractional elliptic problem with the Hardy-Sobolev-Maz'ya potential and critical nonlinearities. By means of variational methods and suitable technique, a positive solution to this problem is obtained. MSC: 35A15; 35B09; 35B33; 35R11

Cited by 2 publications

References 20 publications

A Global multiplicity result for a very singular critical nonlocal equation

A Global multiplicity result for a very singular critical nonlocal equation

Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

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