Pointwise estimates of solutions to semilinear elliptic equations and inequalities

Grigor’yan, Alexander; Verbitsky, Igor E.

doi:10.1007/s11854-019-0004-z

Cited by 10 publications

(16 citation statements)

References 39 publications

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“…By [11,Theorem 3.5] (see also [1], [14]), there exists a positive solutionṽ of the equationṽ = G(ṽ q dσ) +h. It follows thatṽ is a positive solution to (3…”

Section: Preliminariesmentioning

confidence: 99%

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Superlinear elliptic inequalities on manifolds

Grigor’yan

Sun

Verbitsky

2020

Journal of Functional Analysis

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Let M be a complete non-compact Riemannian manifold and let σ be a Radon measure on M . We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliywhere q > 1. We obtain necessary and sufficent criteria for existence of positive solutions in terms of Green function of ∆. In particular, explicit necessary and sufficient conditions are given when M has nonnegative Ricci curvature everywhere in M , or more generally when Green's function satisfies the 3G-inequality.2010 Mathematics Subject Classification. Primary 35J61; Secondary 58J05, 31B10, 42B37.

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“…By [11,Theorem 3.5] (see also [1], [14]), there exists a positive solutionṽ of the equationṽ = G(ṽ q dσ) +h. It follows thatṽ is a positive solution to (3…”

Section: Preliminariesmentioning

confidence: 99%

“…In this section we prove some preparatory results necessary for the proofs of the main theorems. We use some results from [11], [12] and [14].…”

Section: Preliminariesmentioning

confidence: 99%

Superlinear elliptic inequalities on manifolds

Grigor’yan

Sun

Verbitsky

2020

Journal of Functional Analysis

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“…The following theorem provides pointwise estimates for supersolutions to sublinear elliptic equations, see [8,Theorem 1.3].…”

Section: Function Spacesmentioning

confidence: 99%

“…The following pair of iterated pointwise inequalities plays an important role in this paper (see [8,Lemma 2.5]).…”

Section: Function Spacesmentioning

confidence: 99%

Solutions to sublinear elliptic equations with finite generalized energy

Seesanea

Verbitsky

2018

Calc. Var.

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We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation Lu = σu q + µ in Ω, in the sublinear case 0 < q < 1, with finite generalized energy:where γ = 1 corresponds to finite energy solutions.Here Lu := − div(A∇u) is a linear uniformly elliptic operator with bounded measurable coefficients, and σ, µ are nonnegative functions (or Radon measures), on an arbitrary domain Ω ⊆ R n which possesses a positive Green function associated with L.When 0 < γ ≤ 1, this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet spaceẆ 1,p 0 (Ω) for 1 < p ≤ 2.2010 Mathematics Subject Classification. Primary 35J61, 42B37; Secondary 31B10, 31B15.

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“…is well-defined. In [10] it is shown that if q > 0, then u satisfies a pointwise estimate from below, in terms of the functions H andH. On the other hand, if q < 0, then u satisfies a similar pointwise estimate from above.…”

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Pointwise estimates for solutions of semilinear parabolic inequalities with a potential

Montoro

Punzo

Sciunzi

2018

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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We obtain pointwise estimates for solutions of semilinear parabolic equations with a potential on connected domains both of R n and of general Riemannian manifolds.2010 Mathematics Subject Classification: 35K55,35K08,35K01 IntroductionWe are concerned with solutions of semilinear parabolic equations of the following type:( 1.1) where Ω ⊆ M is a connected domain on a complete Riemannian manifold, the potential V = V (x, t) and the source term f = f (x, t) are given continuous functions in Q T . Moreover, we suppose that f ≥ 0, f ≡ 0, while V can be signed. We consider both the case q > 0 and u ≥ 0, and that q < 0 and u > 0. The elliptic counterpart of equation (1.1), that is − ∆u + V u q = f in Ω, (1.2)

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Pointwise estimates of solutions to semilinear elliptic equations and inequalities

Abstract: We study pointwise behavior of positive solutions to nonlinear integral equations, and related inequalities, of the type

Cited by 10 publications

References 39 publications

Superlinear elliptic inequalities on manifolds

Superlinear elliptic inequalities on manifolds

Solutions to sublinear elliptic equations with finite generalized energy

Pointwise estimates for solutions of semilinear parabolic inequalities with a potential

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