Superlinear elliptic inequalities on manifolds

Grigor’yan, Alexander; Sun, Yan‐Qiong; Verbitsky, Igor E.

doi:10.1016/j.jfa.2019.108444

Cited by 10 publications

(8 citation statements)

References 20 publications

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“…Recently in the paper [15], Grigor'yan, Verbitsky and the first author proved that on the manifold where the above mentioned conditions (VD) and (PI) are both satisfied, then (1.11) possesses a C 2 positive solution if and only if…”

Section: Introductionmentioning

confidence: 99%

Liouville's theorems to quasilinear differential inequalities involving gradient nonlinearity term on manifolds

Sun¹,

Xu²

2021

Preprint

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We investigate the nonexistence and existence of nontrivial positive solutions to ∆mu + u p |∇u| q ≤ 0 on noncompact geodesically complete Riemannian manifolds, where m > 1, and (p, q) ∈ R 2 . According to classification of (p, q), we establish different volume growth conditions to obtain Liouville's theorems for the above quasilinear differential inequalities, and we also show these volume growth conditions are sharp in most cases. Moreover, the results are completely new for (p, q) of negative pair, even in the Euclidean space.

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Section: Introductionmentioning

confidence: 99%

Liouville's theorems to quasilinear differential inequalities involving gradient nonlinearity term on manifolds

Sun¹,

Xu²

2021

Preprint

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“…We do not know whether condition (p 0 ) can be removed. The following conjecture is motivated by [11,Conjecture 1].…”

Section: Introductionmentioning

confidence: 99%

Superlinear elliptic inequalities on weighted graphs

Gu¹,

Huang²,

Sun³

2022

Preprint

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Let (V, µ) be an infinite, connected, locally finite weighted graph. We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality ∆u + u σ ≤ 0 in V, where ∆ is the standard graph Laplacian on V and σ > 0. For σ ∈ (0, 1], the inequality admits no nontrivial positive solution. For σ > 1, assuming condition (p0) on (V, µ), we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that isfor some o ∈ V and all large enough n. For any ε > 0, we can construct an example on a homogeneous tree TN with µ(o, n) ≈ n 2σ σ−1 (ln n) 1 σ−1 +ε , and a solution to the inequality on (TN , µ) to illustrate the sharpness of 2σ σ−1 and 1 σ−1 .

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“…Quasi-metric kernels have numerous applications in Analysis and PDE, including weighted norm inequalities, Schrödinger operators and 3-G inequalities, spectral theory, semilinear elliptic problems on R n and complete, non-compact Riemannian manifolds, etc. (see, for instance, [3], [10], [17], [19], [20], [22], [27], [28]).…”

Section: Introductionmentioning

confidence: 99%

“…Let M be a complete, non-compact Riemannian manifold with the volume doubling condition. If the minimal Green's function G satisfies the Li-Yau estimates, then G is known to be a quasi-metric kernel [17,Lemma 6.1]. In particular, this is true on manifolds M with nonnegative Ricci curvature, and in many other circumstances (see [17]).…”

Section: Introductionmentioning

confidence: 99%

Global pointwise estimates of positive solutions to sublinear equations

Verbitsky¹

2022

Preprint

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We give bilateral pointwise estimates for positive solutions u to the sublinear integral equation u = G(σu q ) + f in Ω, for 0 < q < 1, where σ ≥ 0 is a measurable function, or a Radon measure, f ≥ 0, and G is the integral operator associated with a positive kernel G on Ω × Ω. Our main results, which include the existence criteria and uniqueness of solutions, hold for quasimetric, or quasi-metrically modifiable kernels G.As a consequence, we obtain bilateral estimates, along with the existence and uniqueness, for positive solutions u, possibly unbounded, to sublinear elliptic equations involving the fractional Laplacian,where 0 < q < 1, and µ, σ ≥ 0 are measurable functions, or Radon measures, on a bounded uniform domain Ω ⊂ R n for 0 < α ≤ 2, or on the entire space R n , a ball or half-space, for 0 < α < n. Contents 1. Introduction 2 2. Kernels and potential theory 11 3. Lower bounds for supersolutions 14 4. Quasi-metric kernels 17 5. Quasi-metrically modifiable kernels 25 6. Proofs of Theorem 1.1, Theorem 1.2, and Theorem 6.1 30 References 33

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Superlinear elliptic inequalities on manifolds

Cited by 10 publications

References 20 publications

Liouville's theorems to quasilinear differential inequalities involving gradient nonlinearity term on manifolds

Liouville's theorems to quasilinear differential inequalities involving gradient nonlinearity term on manifolds

Superlinear elliptic inequalities on weighted graphs

Global pointwise estimates of positive solutions to sublinear equations

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