Quasilinear elliptic equations and weighted Sobolev–Poincaré inequalities with distributional weights

Jaye, Benjamin; Maz′ya, Vladimir; Verbitsky, Igor E.

doi:10.1016/j.aim.2012.09.029

Cited by 22 publications

(28 citation statements)

References 23 publications

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“…As far as existence is concerned, the nonlinearity

{false| \nabla u false|}^{p}

in is considered ‘to have the bad sign’ and by now it is well known that in order for to have a solution the datum

σ

must be both small and regular enough . In particular, if

σ

is a nonnegative distribution in

normalΩ

(that is, a nonnegative locally finite measure in

normalΩ

), then a necessary condition for the first equation in to have a

W_{loc}^{1, p} false(normalΩ false)

solution is that (see )

0true \int_{Ω} {false| φ false|}^{p} d σ ⩽ λ \int_{Ω} {false| \nabla φ false|}^{p} d x for all φ \in C_{c}^{\infty} (Ω),

with

λ = {(p - 1)}^{p - 1}

. Moreover, when

σ ⩾ 0

the nonlinear term itself also obeys a similar Poincaré–Sobolev inequality

0true \int_{Ω} {false| φ false|}^{p} {false| \nabla u false|}^{p} d x ⩽ A \int_{Ω} {false| \nabla φ false|}^{p} d x for all φ \in C_{c}^{\infty} (Ω),

with

A = p^{p}

.…”

Section: Introductionmentioning

confidence: 99%

“…Remark If the factor

{false(p - 1 false)}^{1 - p}

on the right‐hand side of is dropped, then the smallness condition on

λ

becomes

λ \in (0, p^{#})

, where

p^{#} = {(p - 1)}^{2 - p}

p > 2

and

p^{#} = 1

p ⩽ 2

as in . The sharpness of

p^{#}

(and that of

{false(p - 1 false)}^{trueprefixmin false{1, p - 1 false}}

for ) was also justified in .…”

Section: Introductionmentioning

confidence: 99%

“…We mention that the existence of finite energy solutions to in the case

σ ⩾ 0

was obtained in by a method that does not seem to work for sign changing

σ

(see also for

p = 2

). On the other hand, the work (see also ) obtains a locally finite energy solution

v \in W_{loc}^{1, p} false(normalΩ false)

to the first two equations in ( without any boundary conditions ) only under the mild restriction

0true - normalΛ \int_{Ω} {false| \nabla φ false|}^{p} d x ⩽ {false⟨ σ, false| φ false|}^{p} ⟩ ⩽ λ \int_{Ω} {false| \nabla φ false|}^{p} d x for all φ \in C_{c}^{\infty} (Ω)

for some

λ \in (0, {(p - 1)}^{min {1, p - 1}})

and

Λ \in (0, + \infty)

. Moreover,

v

also satisfies for some

A > 0

.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover,

v

also satisfies for some

A > 0

. Then, also under the restriction

λ \in (0, {(p - 1)}^{min {1, p - 1}})

, by the logarithmic transformation

u = (p - 1) log (v)

, it was obtained in a solution

u \in W_{loc}^{1, p} false(normalΩ false)

to the first equation in (but without any boundary condition) that also satisfies for some

A > 0

.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation

Adimurthi

Phuc

2018

J. London Math. Soc.

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We obtain necessary and sufficient conditions with sharp constants on the distribution σ for the existence of a globally finite energy solution to the quasi‐linear equation with a gradient source term of natural growth of the form −normalΔpu=|∇u|p+σ in a bounded open set Ω⊂double-struckRn. Here Δp, p>1, is the standard p‐Laplacian operator defined by normalΔpu= div (|∇u|p−2∇ufalse). The class of solutions that we are interested in consists of functions u∈W01,pfalse(normalΩfalse) such that eμu∈W01,pfalse(normalΩfalse) for some μ>0 and the inequality 0true∫Ωfalse|φfalse|pfalse|∇ufalse|pdx⩽A∫Ωfalse|∇φfalse|pdxholds for all φ∈Cc∞false(normalΩfalse) with some constant A>0. This is a natural class of solutions at least when the distribution σ is nonnegative. The study of −normalΔpu=|∇u|p+σ is applied to show the existence of globally finite energy solutions to the quasi‐linear equation of Schrödinger type −normalΔpv=σvp−1, v⩾0 in normalΩ, and v=1 on ∂Ω, via the exponential transformation u↦v=eup−1.

show abstract

“…As far as existence is concerned, the nonlinearity

{false| \nabla u false|}^{p}

in is considered ‘to have the bad sign’ and by now it is well known that in order for to have a solution the datum

σ

must be both small and regular enough . In particular, if

σ

is a nonnegative distribution in

normalΩ

(that is, a nonnegative locally finite measure in

normalΩ

), then a necessary condition for the first equation in to have a

W_{loc}^{1, p} false(normalΩ false)

solution is that (see )

0true \int_{Ω} {false| φ false|}^{p} d σ ⩽ λ \int_{Ω} {false| \nabla φ false|}^{p} d x for all φ \in C_{c}^{\infty} (Ω),

with

λ = {(p - 1)}^{p - 1}

. Moreover, when

σ ⩾ 0

the nonlinear term itself also obeys a similar Poincaré–Sobolev inequality

0true \int_{Ω} {false| φ false|}^{p} {false| \nabla u false|}^{p} d x ⩽ A \int_{Ω} {false| \nabla φ false|}^{p} d x for all φ \in C_{c}^{\infty} (Ω),

with

A = p^{p}

.…”

Section: Introductionmentioning

confidence: 99%

“…Remark If the factor

{false(p - 1 false)}^{1 - p}

on the right‐hand side of is dropped, then the smallness condition on

λ

becomes

λ \in (0, p^{#})

, where

p^{#} = {(p - 1)}^{2 - p}

p > 2

and

p^{#} = 1

p ⩽ 2

as in . The sharpness of

p^{#}

(and that of

{false(p - 1 false)}^{trueprefixmin false{1, p - 1 false}}

for ) was also justified in .…”

Section: Introductionmentioning

confidence: 99%

“…We mention that the existence of finite energy solutions to in the case

σ ⩾ 0

was obtained in by a method that does not seem to work for sign changing

σ

(see also for

p = 2

). On the other hand, the work (see also ) obtains a locally finite energy solution

v \in W_{loc}^{1, p} false(normalΩ false)

to the first two equations in ( without any boundary conditions ) only under the mild restriction

0true - normalΛ \int_{Ω} {false| \nabla φ false|}^{p} d x ⩽ {false⟨ σ, false| φ false|}^{p} ⟩ ⩽ λ \int_{Ω} {false| \nabla φ false|}^{p} d x for all φ \in C_{c}^{\infty} (Ω)

for some

λ \in (0, {(p - 1)}^{min {1, p - 1}})

and

Λ \in (0, + \infty)

. Moreover,

v

also satisfies for some

A > 0

.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover,

v

also satisfies for some

A > 0

. Then, also under the restriction

λ \in (0, {(p - 1)}^{min {1, p - 1}})

, by the logarithmic transformation

u = (p - 1) log (v)

, it was obtained in a solution

u \in W_{loc}^{1, p} false(normalΩ false)

to the first equation in (but without any boundary condition) that also satisfies for some

A > 0

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation

Adimurthi

Phuc

2018

J. London Math. Soc.

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“…Let V ∈ D admits a positive solution (in a certain weak sense) for any α ∈ (0, p ♯ ), where p ♯ < 1 is given explicitly and depends only on p (see [21,Theorem 1.2 (i)], or [20, Theorem 1.1 (i)] for p = 2). Moreover, this range for α is optimal as examples involving the Hardy potential reveals (see [21,Remark 1.3], or [20,Example 7.3] for p = 2). We note that under the above assumptions, the local Harnack inequality for positive solutions of (1.5) is in general not valid.…”

Section: Introductionmentioning

confidence: 99%

On positive solutions of the (p,A)-Laplacian with potential in Morrey space

Pinchover

Psaradakis

2016

Anal. PDE

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We study qualitative positivity properties of quasilinear equations of the formwhere Ω is a domain inis a symmetric and locally uniformly positive definite matrix, V is a real potential in a certain local Morrey space (depending on p), andOur assumptions on the coefficients of the operator for p ≥ 2 are the minimal (in the Morrey scale) that ensure the validity of the local Harnack inequality and hence the Hölder continuity of the solutions. For some of the results of the paper we need slightly stronger assumptions when p < 2.We prove an Allegretto-Piepenbrink-type theorem for the operator Q ′ A,p,V , and extend criticality theory to our setting. Moreover, we establish a Liouville-type theorem and obtain some perturbation results. Also, in the case 1 < p ≤ n, we examine the behavior of a positive solution near a nonremovable isolated singularity and characterize the existence of the positive minimal Green function for the operator Q ′ A,p,V [u] in Ω.

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Quasilinear elliptic equations with exponential nonlinearity and measure data

Huang

2020

Math Methods in App Sciences

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Let normalΩ⊆double-struckRN be either a bounded domain or the entire space. This paper concerns existence and global pointwise estimates in terms of Wolff potentials of solutions to quasilinear and Hessian equations of Lane‐Emden type: −Δpu=σP(u)+ω,Fk[−u]=σP(u)+ω under minimal assumptions on σ and ω, where σ is an A∞ weight and ω is a nonnegative Borel measure, and Pfalse(ufalse)∼expγuβ with γ>0, β ≥ 1. Our methods are based on systematic application of Wolff potentials and good‐λ inequality for fractional maximal potential.

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Quasilinear elliptic equations and weighted Sobolev–Poincaré inequalities with distributional weights

Cited by 22 publications

References 23 publications

Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation

Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation

On positive solutions of the (p,A)-Laplacian with potential in Morrey space

Quasilinear elliptic equations with exponential nonlinearity and measure data

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