Nonlocal problems with singular nonlinearity

Canino, Annamaria; Montoro, Luigi; Sciunzi, Berardino; Squassina, Marco

doi:10.1016/j.bulsci.2017.01.002

Cited by 74 publications

(76 citation statements)

References 23 publications

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“…The same holds true if

α > 1

and

ω \in L^{1} false(normalΩ false)

. Indeed, in [, Lemma 3.4] the authors proved that, under these hypotheses, the sequence

{\{u_{n}^{false(α - 1 + p false) / p}\}}_{n \in double-struckN}

is bounded in

W_{0}^{s, p} false(normalΩ false)

. This fact and the monotonicity of

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

imply that

u_{α}^{(α - 1 + p) / p} \in L^{1} false(normalΩ false)

, so that

u_{α} false(x false) < \infty

for almost every

x \in Ω

.…”

Section: Preliminariesmentioning

confidence: 87%

“…Proof The first inequality in is proved in . Thus, by using this monotonicity and

φ = u_{n + 1}

in we obtain the second inequality:

{([], u_{n})}_{s, p}^{p} \leq {([], u_{n + 1})}_{s, p}^{p} + p \int_{Ω} \frac{u_{n} - u_{n + 1}}{{(u_{n} + \frac{1}{n})}^{α}} ω_{n} d x \leq {([], u_{n + 1})}_{s, p}^{p} .

Let φ be any nonnegative function in

W_{0}^{s, p} false(normalΩ false)

.…”

Section: Preliminariesmentioning

confidence: 94%

“…Proof Adapting the script developed in , the function

u_{n}

is obtained in [, Proposition 2.3] as a fixed point of the operator

T : L^{p} false(normalΩ false) ⟶ W_{0}^{s, p} false(normalΩ false) ↪ L^{p} false(normalΩ false)

that assigns to each

w \in L^{p} false(normalΩ false)

the only weak solution

v = T false(w false) \in W_{0}^{s, p} false(normalΩ false)

of the nonsingular Dirichlet problem

({, \begin{matrix} {(- Δ_{p})}^{s} u = \frac{ω_{n}}{{(() w_{+} + \frac{1}{n})}^{α}} & in & Ω, \\ u = 0 & in & {double-struckR}^{N} ∖ Ω . \end{matrix})

Since the right‐hand term of the above equation is nonnegative and belongs to

L^{\infty} false(normalΩ false)

, we can apply the comparison principle for the fractional p ‐Laplacian (see [, Lemma 9]) and the main result of to conclude, respectively, that

v = T (w) \geq 0

almost everywhere in Ω and belongs to

C^{β_{s}} true(\bar{Ω} true)

for some

β_{s} \in false(0, s false]

, which does neither depend on α nor on n . Of course, the fixed point

u_{n}

also enjoys these properties.…”

Section: Preliminariesmentioning

confidence: 99%

“…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%

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Fractional Sobolev inequalities associated with singular problems

Ercole

Pereira

2018

Mathematische Nachrichten

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show abstract

“…The same holds true if

α > 1

and

ω \in L^{1} false(normalΩ false)

. Indeed, in [, Lemma 3.4] the authors proved that, under these hypotheses, the sequence

{\{u_{n}^{false(α - 1 + p false) / p}\}}_{n \in double-struckN}

is bounded in

W_{0}^{s, p} false(normalΩ false)

. This fact and the monotonicity of

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

imply that

u_{α}^{(α - 1 + p) / p} \in L^{1} false(normalΩ false)

, so that

u_{α} false(x false) < \infty

for almost every

x \in Ω

.…”

Section: Preliminariesmentioning

confidence: 87%

“…Proof The first inequality in is proved in . Thus, by using this monotonicity and

φ = u_{n + 1}

in we obtain the second inequality:

{([], u_{n})}_{s, p}^{p} \leq {([], u_{n + 1})}_{s, p}^{p} + p \int_{Ω} \frac{u_{n} - u_{n + 1}}{{(u_{n} + \frac{1}{n})}^{α}} ω_{n} d x \leq {([], u_{n + 1})}_{s, p}^{p} .

Let φ be any nonnegative function in

W_{0}^{s, p} false(normalΩ false)

.…”

Section: Preliminariesmentioning

confidence: 94%

“…Proof Adapting the script developed in , the function

u_{n}

is obtained in [, Proposition 2.3] as a fixed point of the operator

T : L^{p} false(normalΩ false) ⟶ W_{0}^{s, p} false(normalΩ false) ↪ L^{p} false(normalΩ false)

that assigns to each

w \in L^{p} false(normalΩ false)

the only weak solution

v = T false(w false) \in W_{0}^{s, p} false(normalΩ false)

of the nonsingular Dirichlet problem

({, \begin{matrix} {(- Δ_{p})}^{s} u = \frac{ω_{n}}{{(() w_{+} + \frac{1}{n})}^{α}} & in & Ω, \\ u = 0 & in & {double-struckR}^{N} ∖ Ω . \end{matrix})

Since the right‐hand term of the above equation is nonnegative and belongs to

L^{\infty} false(normalΩ false)

, we can apply the comparison principle for the fractional p ‐Laplacian (see [, Lemma 9]) and the main result of to conclude, respectively, that

v = T (w) \geq 0

almost everywhere in Ω and belongs to

C^{β_{s}} true(\bar{Ω} true)

for some

β_{s} \in false(0, s false]

, which does neither depend on α nor on n . Of course, the fixed point

u_{n}

also enjoys these properties.…”

Section: Preliminariesmentioning

confidence: 99%

“…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%

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Fractional Sobolev inequalities associated with singular problems

Ercole

Pereira

2018

Mathematische Nachrichten

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On a degenerate singular elliptic problem

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Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

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Math Methods in App Sciences

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Nonlocal problems with singular nonlinearity

Abstract: Abstract. We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian operator and singular nonlinearities.

Cited by 74 publications

References 23 publications

Fractional Sobolev inequalities associated with singular problems

Fractional Sobolev inequalities associated with singular problems

On a degenerate singular elliptic problem

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

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