Infinitely many positive solutions for a nonlocal problem

Gu, Guangze; Zhang, Wei; Zhao, Fukun

doi:10.1016/j.aml.2018.04.010

Cited by 11 publications

(5 citation statements)

References 19 publications

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“…This is bad news as far as the application of theorems in the variational analysis is concerned because, most of them demand the energy functional to be C 1 over its domain of definition. Thus, inspired by the cutoff technique introduced by Clark and Gilbarg [12] and later used by Gu et al [18], we shall apply it to problem (3.1).…”

Section: Weak Solutionsmentioning

confidence: 99%

On critical variable-order Kirchhoff type problems with variable singular exponent

Zuo,

Choudhuri,

Repovš

2022

Preprint

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We establish a continuous embedding W s(•),2 (Ω) ֒→ L α(•) (Ω), where the variable exponent α(x) can be close to the critical exponent 2, with s(x) = s(x, x) for all x ∈ Ω. Subsequently, this continuous embedding is used to prove the multiplicity of solutions for critical nonlocal degenerate Kirchhoff problems with a variable singular exponent. Moreover, we also obtain the uniform L ∞ -estimate of these infinite solutions by a bootstrap argument.

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Section: Weak Solutionsmentioning

confidence: 99%

On critical variable-order Kirchhoff type problems with variable singular exponent

Zuo,

Choudhuri,

Repovš

2022

Preprint

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“…This led to a chain of works that can be found in [3,4,5,6] and the references therein. Meanwhile, another noteworthy development took place due to Lazer and McKenna [7] in studying elliptic partial differential equations involving a singularity.…”

Section: Introductionmentioning

confidence: 99%

Infinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity

Ghosh,

Choudhuri,

Giri

2020

Preprint

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We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularitieswhere Ω ⊂ R N , N ≥ 2 is a smooth, bounded domain, λ > 0, s ∈ (0, 1), γ(x) ∈ (0, 1) for all x ∈ Ω, N > sp(x, y) for all (x, y) ∈ Ω × Ω and (−∆) s p(•) is the fractional p(•)-Laplacian operator with variable exponent. The nonlinear function f satisfies certain growth conditions. Moreover, we establish a uniform L ∞ ( Ω) estimate of the solution(s) by the Moser iteration technique.

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“…Recently, Gu et al [18] has guaranteed the existence of infinitely many solutions to a nonlocal problem of the following type with a sublinear growth of f .…”

Section: Introductionmentioning

confidence: 99%

“…To our knowledge, the study of the problem (1.1) is very new to the literature due to the presence of a singular term u −γ . Motivated from [18], we will prove the existence of infinitely many weak solutions to the problem (1.1). We assume the following growth conditions on f .…”

Section: Introductionmentioning

confidence: 99%

Existence of infinitely many solutions for a nonlocal elliptic PDE involving singularity

Ghosh

Choudhuri

2019

Positivity

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Infinitely many positive solutions for a nonlocal problem

Cited by 11 publications

References 19 publications

On critical variable-order Kirchhoff type problems with variable singular exponent

On critical variable-order Kirchhoff type problems with variable singular exponent

Infinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity

Existence of infinitely many solutions for a nonlocal elliptic PDE involving singularity

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