Local and Global Behaviour of Solutions to Nonlinear Equations with Natural Growth Terms

Jaye, Benjamin; Verbitsky, Igor E.

doi:10.1007/s00205-011-0491-2

Cited by 17 publications

(14 citation statements)

References 39 publications

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“…Similar equations in the supercritical case q > k were considered in [PV08], and in the critical case q = k, in [JV10]. Intrinsic nonlinear potentials of the type K α,p σ do not play a role there.…”

Section: Solutions In L 1+qmentioning

confidence: 76%

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Nonlinear elliptic equations and intrinsic potentials of Wolff type

Cao¹,

Verbitsky²

2017

Journal of Functional Analysis

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Abstract. We give necessary and sufficient conditions for the existence of weak solutions to the model equationin the case 0 < q < p − 1, where σ ≥ 0 is an arbitrary locally integrable function, or measure, and ∆pu = div(∇u|∇u| p−2 ) is the p-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equationwhere b > 0. These results are new even in the classical case p = 2. Our approach is based on the use of special nonlinear potentials of Wolff type adapted for "sublinear" problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators div A(x, ∇u), fractional Laplacians (−∆) α , or fully nonlinear k-Hessian operators.

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Section: Solutions In L 1+qmentioning

confidence: 76%

“…However, the reduction of both (5.5) and (5.7) to (1.17) is carried over as in the case of the p-Laplacian treated above. See details in [PV08], [JV10], [JV12], [CV14].…”

Section: Solutions In L 1+qmentioning

confidence: 99%

Nonlinear elliptic equations and intrinsic potentials of Wolff type

Cao¹,

Verbitsky²

2017

Journal of Functional Analysis

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“…For more related results, one can refer to the previous studies of these and other related results. [11][12][13][14][15][16][17][18][19][20][21][22][23] Before discussing related results of Equation (1) with exponential nonlinearity in details, we would like to recall some definitions. For > 0, p > 1 such that p < N, the R-truncated Wolff potential W R ,p (x) of a nonnegative Borel measure on R N is defined by…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Suppose that (x) is A ∞ weight defined as (41), ∈ + b (R N ) satisfying (7). Then, there exist positive constants c 14 (N, , , r, [ ]…”

Section: Proof Of Theorems 1 and 2: Sufficient Conditionsmentioning

confidence: 99%

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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“…This makes it possible to replace the p-Laplacian ∆ p in the model problem (1.1) by a more general quasilinear operator divA(x, ∇·) with bounded measurable coefficients, under standard structural assumptions on A(x, ξ) which ensure that A(x, ξ) · ξ ≈ |ξ| p [HKM06], [MZ97], or a fully nonlinear operator of k-Hessian type [TW99], [Lab02] (see also [PV09], [JV12]), and treat more general nonlinearities on the right-hand side. Equations involving operators of the p-Laplacian type on Carnot groups can be covered as well using methods developed in [PV13].…”

Section: Introductionmentioning

confidence: 99%

Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms

Dat

Verbitsky

2014

Calc. Var.

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Abstract. We study finite energy solutions to quasilinear elliptic equations of the type −∆pu = σ u q in R n , where ∆p is the p-Laplacian, p > 1, and σ is a nonnegative function (or measure) on R n , in the case 0 < q < p − 1 (below the "natural growth" rate q = p − 1). We give an explicit necessary and sufficient condition on σ which ensures that there exists a solution u in the homogeneous Sobolev space L 1,p 0 (R n ), and prove its uniqueness. Among our main tools are integral inequalities closely associated with this problem, and Wolff potential estimates used to obtain sharp bounds of solutions. More general quasilinear equations with the A-Laplacian divA(x, ∇·) in place of ∆p are considered as well.

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Local and Global Behaviour of Solutions to Nonlinear Equations with Natural Growth Terms

Cited by 17 publications

References 39 publications

Nonlinear elliptic equations and intrinsic potentials of Wolff type

Nonlinear elliptic equations and intrinsic potentials of Wolff type

Quasilinear elliptic equations with exponential nonlinearity and measure data

Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms

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