Global and local behavior of positive solutions of nonlinear elliptic equations

Gidas, Basilis; Spruck, Joel

doi:10.1002/cpa.3160340406

Cited by 1,141 publications

(834 citation statements)

References 12 publications

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“…Among many results in [15], the Liouville-type theorem for the subcritical elliptic equation is shown. Namely, there is no nontrivial C 2 solution of the problem 9) if N 3 and 1 < p < N+2 N−2 .…”

Section: Introductionmentioning

confidence: 99%

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Wang

Zhu

2012

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Poláčik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.

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Section: Introductionmentioning

confidence: 99%

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Wang

Zhu

2012

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…c) Our proof of Theorem 2.1 relies on nontrivial modifications of the technique developed by Bidaut-Véron [5] for the scalar nonlinear heat equation. The latter was an adaptation of the celebrated method of Gidas and Spruck [12] for elliptic equations (see also [6] for some particular elliptic systems). It is based on nonlinear integral estimates and Bochner formula.…”

Section: Liouville Type Resultsmentioning

confidence: 99%

A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

Phan

Souplet²

2016

Math. Ann.

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Abstract. We prove a Liouville-type theorem for semilinear parabolic systems of the formin the whole space R N × R. Very recently, Quittner [Math. Ann., DOI 10.1007/s00208-015-1219-7 (2015)] has established an optimal result for m = 2 in dimension N ≤ 2, and partial results in higher dimensions in the range p < N/(N − 2). By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron, we partially improve the results of Quittner in dimensions N ≥ 3. In particular, our results solve the important case of the parabolic Gross-Pitaevskii system -i.e. the cubic case r = 1 -in space dimension N = 3, for any symmetric (m, m)-matrix (β ij ) with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouvilletype theorem in the half-space R N + × R. As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.

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“…This is, however, not true in general. In fact, there exist no such metrics on the sphere minus one point, as was proved by Gidas-Ni-Nirenberg [5] and Gidas-Spruck [6]. On the other hand, Schoen [19] constructed such a metric on the complement of any finite set of at least two points on the sphere.…”

Section: The Case μ ι (M) > 0 -Scalar Flat Metricsmentioning

confidence: 95%

Complete conformal metrics with prescribed scalar curvature on subdomains of a compact manifold

Kato

Nayatani

1993

Nagoya Mathematical Journal

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Let (M, g) be a Riemannian manifold of dimension n≥ 3 and ĝanother metric on M which is pointwise conformai to g. It can be written where u is a positive smooth function on M. Then the curvature of g is computable in terms of that of g and the derivatives of u up to second order. In particular, if S and S denote the scalar curvature of g and g respectively, they are related by the equationwhere ▽u denotes the Laplacian of u, defined with respect to the metric g.

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Global and local behavior of positive solutions of nonlinear elliptic equations

Cited by 1,141 publications

References 12 publications

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

Complete conformal metrics with prescribed scalar curvature on subdomains of a compact manifold

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