Semilinear problems for the fractional laplacian with a singular nonlinearity

Barrios, Begoña; Bonis, Ida de; Medina, María; Peral, Ireneo

doi:10.1515/math-2015-0038

Cited by 90 publications

(69 citation statements)

References 36 publications

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“…As for singular problems involving nonlocal (fractional) operators, the literature is quite recent and more restricted to

L u = {(- Δ_{p})}^{s} u

(see ).…”

Section: Introductionmentioning

confidence: 99%

“…Most of the results presented in Section are contained in (for

p = 2

) and (for

p > 1

), but we give some additional contribution. For example, by observing that

{[u_{n}]}_{s, p}^{p} \leq {[φ]}_{s, p}^{p} + p \int_{Ω} \frac{u_{n} - φ}{{(u_{n} + \frac{1}{n})}^{α}} ω_{n} d x, for all φ \in W_{0}^{s, p} false(normalΩ false),

we verify that

{[u_{n}]}_{s, p} \leq {[u_{n + 1}]}_{s, p}

for all

n \in N

and present a simpler proof that

u_{n}

converges strongly to a solution of provided that

{\{u_{n}\}}_{n \in double-struckN}

is bounded in

W_{0}^{s, p} false(normalΩ false)

(see Proposition ).…”

Section: Introductionmentioning

confidence: 99%

“…The literature on singular problems for equations of the form  = − has primarily focused on local operators as the Laplacian,  = − div ∇ (see [2,4,7,10,17,18,21]), or the -Laplacian,  = − div ( |∇ | −2 ∇ ) , > 1 (see [1,9,[13][14][15]20]). As for singular problems involving nonlocal (fractional) operators, the literature is quite recent and more restricted to  = ( −Δ ) (see [3,8]). In general, the energy functional associated with a singular problem of the form  = − is not differentiable.…”

Section: Introductionmentioning

confidence: 99%

“…Most of the results presented in Section 2 are contained in [3] (for = 2) and [8] (for > 1), but we give some additional contribution. For example, by observing that…”

Section: Introductionmentioning

confidence: 99%

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Fractional Sobolev inequalities associated with singular problems

Ercole

Pereira

2018

Mathematische Nachrichten

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“…As for singular problems involving nonlocal (fractional) operators, the literature is quite recent and more restricted to

L u = {(- Δ_{p})}^{s} u

(see ).…”

Section: Introductionmentioning

confidence: 99%

“…Most of the results presented in Section are contained in (for

p = 2

) and (for

p > 1

), but we give some additional contribution. For example, by observing that

{[u_{n}]}_{s, p}^{p} \leq {[φ]}_{s, p}^{p} + p \int_{Ω} \frac{u_{n} - φ}{{(u_{n} + \frac{1}{n})}^{α}} ω_{n} d x, for all φ \in W_{0}^{s, p} false(normalΩ false),

we verify that

{[u_{n}]}_{s, p} \leq {[u_{n + 1}]}_{s, p}

for all

n \in N

and present a simpler proof that

u_{n}

converges strongly to a solution of provided that

{\{u_{n}\}}_{n \in double-struckN}

is bounded in

W_{0}^{s, p} false(normalΩ false)

(see Proposition ).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Most of the results presented in Section 2 are contained in [3] (for = 2) and [8] (for > 1), but we give some additional contribution. For example, by observing that…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fractional Sobolev inequalities associated with singular problems

Ercole

Pereira

2018

Mathematische Nachrichten

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“…Problems of the type (1.1) arise in certain problems in fluid mechanics, pseudoplastic flow, chemical heterogenous catalysts and non-Newtonian fluids. The topic of the existence of weak solutions to problem (1.1) has been studied extensively in [2,6,8,11] and the references there in. In [11], authors proved the existence and multiplicity results on problem (1.4), where 0 < γ 1.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian

Wang¹,

Zhang²

2018

J. Nonlinear Sci. Appl.

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In this paper we consider the existence and uniqueness of weak positive solution for nonlocal equations of the square root of the Laplacian with singular nonlinearity. The remarkable feature of this paper is the fact that the natural associated functional fails to be Fréchet differentiable, critical point theory could not be applied to obtain the existence of weak positive solution. We first establish the priori estimate of weak solution of approximating problems. Then the weak positive solution is constructed by combining sub-and supersolutions method and truncate technology.

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A study of extremal parameter for fractional singular Choquard problem

Mishra

Tripathi

2023

Mathematische Nachrichten

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In this work, we study the singular problem involving fractional Laplacian operator perturbed with a Choquard nonlinearity using the idea of constrained minimization based on Nehari manifold. Precisely, for some 𝜖 > 0, we have proved the existence of two solutions when the parameter 𝜆 ∈ (0, 𝜆 * + 𝜖), adding to the existing works dealing with multiplicity of solutions when the parameter 𝜆 strictly lies below 𝜆 * . We have given a variational characterization of the parametric value 𝜆 * , which is an extremal value of the parameter 𝜆 involved in the problem up to which the Nehari manifold method can be applied successfully. The paper highlights a fine analysis via fibering maps even for 𝜆 ≥ 𝜆 * to establish an existence of two different positive solutions for the underlying problem.

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Semilinear problems for the fractional laplacian with a singular nonlinearity

Cited by 90 publications

References 36 publications

Fractional Sobolev inequalities associated with singular problems

Fractional Sobolev inequalities associated with singular problems

Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian

A study of extremal parameter for fractional singular Choquard problem

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