Liouville type theorems and periodic solutions for $χ^{(2)}$ type systems with non-homogeneous nonlinearities

Jevnikar, Aleks; Wang, Jun; Yang, Wen

doi:10.48550/arxiv.2008.13190

Cited by 2 publications

(10 citation statements)

References 30 publications

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“…(ii) The nonexistence of nontrivial solutions of (5.1) in the special case f (s) = s p +s 2 with 1 < p < p B and n ≤ 5 has been proved in [27] (where a class of systems is actually studied), by using the approach from [3]. However, we note that the nonlinearity is asymptotic to a power.…”

Section: Parabolic Liouville Type Theorems Without Scale Invariancementioning

confidence: 97%

Universal estimates and Liouville theorems for superlinear problems without scale invariance

Souplet

2022

Preprint

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We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale invariant even asymptotically and whose behavior can be quite far from power like.In this enlarged framework, by adapting the doubling-rescaling method from [37,38], we show that the equivalence found there between universal estimates and Liouville theorems remains valid. In the parabolic case we also prove a Liouville type theorem for a rather large class of non scale invariant nonlinearities. This leads to a number of new results for non scale invariant elliptic and parabolic problems, concerning space or space-time singularity estimates, initial and final blow-up rates, universal and a priori bounds for global solutions, and decay rates in space and/or time.We illustrate our approach by a number of examples, which in turn give indication about the optimality of the estimates and of the assumptions. Contents 15 9. Proof of Theorems 6 and 7 23 10. Technical lemmas 30 References 34

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Section: Parabolic Liouville Type Theorems Without Scale Invariancementioning

confidence: 97%

Universal estimates and Liouville theorems for superlinear problems without scale invariance

Souplet

2022

Preprint

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“…(i) Sharpness of the assumptions. Observe that the "homogeneous" estimate (27) implies 7 the Liouville property in Q. Consequently (27) in particular fails whenever (8) admits a positive solution. In this respect we have the following useful counter-example.…”

Section: 1mentioning

confidence: 99%

“…(ii) Decay of bounded solutions. If one considers bounded solutions of ( 19) in an unbounded domain D and is only interested in the decay rates as |x| and/or |t| → ∞, then the assumptions on f for large s (namely s ≥ M = u ∞ ) are not needed, and estimate (27) remains true (with a constant C depending also on M ). 8 We also have an analogue of Theorem 4.1 for boundary value problems, for which we define the additional exponent p * * = n+1 n−1 in case (10) and p * * = 1 + 2 n+1 in case (11).…”

Section: 1mentioning

confidence: 99%

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Universal estimates and Liouville theorems for superlinear problems without scale invariance

Souplet¹

2023

DCDS

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We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale-invariant even asymptotically and whose behavior can be quite far from power like.In this enlarged framework, by adapting the doubling-rescaling method from [<xref ref-type="bibr" rid="b37">37</xref>, <xref ref-type="bibr" rid="b38">38</xref>], we show that the equivalence found there between universal estimates and Liouville theorems remains valid. In the parabolic case we also prove a Liouville type theorem for a rather large class of non scale-invariant nonlinearities. This leads to a number of new results for non scale-invariant elliptic and parabolic problems, concerning space or space-time singularity estimates, initial and final blow-up rates, universal and a priori bounds for global solutions, and decay rates in space and/or time.We illustrate our approach by a number of examples, which in turn give indication about the optimality of the estimates and of the assumptions.

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Liouville type theorems and periodic solutions for $χ^{(2)}$ type systems with non-homogeneous nonlinearities

Cited by 2 publications

References 30 publications

Universal estimates and Liouville theorems for superlinear problems without scale invariance

Universal estimates and Liouville theorems for superlinear problems without scale invariance

Universal estimates and Liouville theorems for superlinear problems without scale invariance

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