Pointwise estimates of Brezis–Kamin type for solutions of sublinear elliptic equations

Cao, Dat; Verbitsky, Igor E.

doi:10.1016/j.na.2016.08.008

Cited by 19 publications

(20 citation statements)

References 21 publications

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“…Unfortunately, behavior of K 1,p,q σ can not be easily calculated from its definition. Cao and Verbitsky [11] constructed weak solutions in W 1,p loc (R n ) under a certain capacity condition and gave two-sided pointwise estimates of such solutions. Seesanea and Verbitsky [23] gave a sufficient condition for the existence of L r -integrable p-superharmonic solutions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms

Hara

Seesanea

2020

Nonlinear Analysis

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We study the existence of positive solutions to quasilinear elliptic equations of the typein the sub-natural growth case 0 < q < p − 1, where ∆ p u = ∇ · (|∇u| p−2 ∇u) is the p-Laplacian with 1 < p < n, and σ and µ are nonnegative Radon measures on R n . We construct minimal generalized solutions under certain generalized energy conditions on σ and µ. To prove this, we give new estimates for interaction between measures. We also construct solutions to equations with several sub-natural growth terms using the same methods.✩ This is a pre-print of an article published in Nonlinear Analysis. The final authenticated version is available online at: https://doi.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms

Hara

Seesanea

2020

Nonlinear Analysis

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“…under additional assumptions on σ, can be found in [9]. As was pointed out in [6], both the lower and upper estimates in (3.5) are sharp in a sense.…”

Section: Matching Upper and Lower Pointwise Estimates Of Solutionsmentioning

confidence: 65%

“…This estimate was obtained for the Laplacian = Δ in the case Ω = R in [6], without specifying the sharp constant (1 − ) 1 1− , under some additional assumptions on σ ≥ 0 (see also [9]). …”

Section: Theorem 22 Let ω ⊆ R Be a Domain With A Positive Green Funmentioning

confidence: 99%

“…Condition α σ < ∞ a.e. alone is not enough to ensure the existence of a global solution even if all the local embedding constants κ( ) in (3.6) are finite, unless σ is radially symmetric (see [9]). …”

Section: Matching Upper and Lower Pointwise Estimates Of Solutionsmentioning

confidence: 99%

“…For entire solutions on Ω = R and = −σ ≥ 0, where σ is a locally finite positive Borel measure in R , there are more complicated matching upper and lower bounds given in terms of certain nonlinear potentials of Th. Wolff type [8][9][10]. Earlier pointwise estimates for bounded positive solutions are due to Brezis and Kamin [6].…”

Section: Introductionmentioning

confidence: 99%

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Pointwise Estimates of Solutions and Existence Criteria for Sublinear Elliptic Equations

Verbitsky¹

2017

Vestn. Volgogr. gos. univ., Ser. 1. Math. Phy. and Comp. Model.

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Abstract. We give a survey of recent results on positive solutions to sublinear elliptic equations of the type − + = , where is an elliptic operator in divergence form, 0 < < 1, ≥ 0 and is a function that may change sign, in a domain Ω ⊆ R , or in a weighted Riemannian manifold, with a positive Green's function . We discuss the existence, as well as global lower and upper pointwise estimates of classical and weak solutions , and conditions that ensure ∈ (Ω) or ∈ 1, (Ω). Some of these results are applicable to homogeneous sublinear integral equations = ( σ) in Ω, where 0 < < 1, and σ = − is a positive locally finite Borel measure in Ω. Here ( σ)( ) = ∫︀ Ω ( , ), ( ) σ( ) is an integral operator with positive (quasi) symmetric kernel on Ω × Ω which satisfies the weak maximum principle. This includes positive solutions, possibly singular, to sublinear equations involving the fractional Laplacian,where 0 < < 1, 0 < α < and = 0 in Ω and at infinity in domains Ω ⊆ R with positive Green's function .

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Weighted Norm Inequalities of (1, q)-Type for Integral and Fractional Maximal Operators

Quinn

Verbitsky

2017

Harmonic Analysis, Partial Differential Equations and Applications

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We study weighted norm inequalities of (1, q)-type for 0 < q < 1,along with their weak-type counterparts, where ν = ν(Ω), and G is an integral operator with nonnegative kernel,These problems are motivated by sublinear elliptic equations in a domain Ω ⊂ R n with non-trivial Green's function G(x, y) associated with the Laplacian, fractional Laplacian, or more general elliptic operator.We also treat fractional maximal operators M α (0 ≤ α < n) on R n , and characterize strong-and weak-type (1, q)-inequalities for M α and more general maximal operators, as well as (1, q)-Carleson measure inequalities for Poisson integrals.2010 Mathematics Subject Classification. Primary 35J61, 42B37; Secondary 31B15, 42B25.

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Pointwise estimates of Brezis–Kamin type for solutions of sublinear elliptic equations

Cited by 19 publications

References 21 publications

Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms

Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms

Pointwise Estimates of Solutions and Existence Criteria for Sublinear Elliptic Equations

Weighted Norm Inequalities of (1, q)-Type for Integral and Fractional Maximal Operators

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