2015
DOI: 10.1090/proc/12647
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Rigidity results for stable solutions of symmetric systems

Abstract: Abstract. We study stable solutions of the following nonlinear systemand Ω is a domain in R n . We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of H is symmetric. It seems that this concept is crucial to prove Liouville theorems, when Ω = R n , and regularity results, when Ω = B 1 , for stable solutions of the above system for a general nonlinearity H ∈ C 1 (R m ). Moreover, we provide an improvement for a linear Liouville… Show more

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Cited by 15 publications
(28 citation statements)
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“…We are now ready to prove the stability inequality for solutions of (1.15). Note that such an inequality for the case of semilinear systems is given in [20,34,35].…”
Section: Geometric Poincaré and Stability Inequalities For Systemsmentioning
confidence: 99%
See 1 more Smart Citation
“…We are now ready to prove the stability inequality for solutions of (1.15). Note that such an inequality for the case of semilinear systems is given in [20,34,35].…”
Section: Geometric Poincaré and Stability Inequalities For Systemsmentioning
confidence: 99%
“…Throughout this paper we use the notation u = (u i ) m i=1 , H(u) = (H i (u)) m i=1 and ∂ j H i (u) = ∂Hi(u) ∂uj . We assume that ∂ i H j (u)∂ j H i (u) > 0 for 1 ≤ i ≤ j ≤ m. The next definition is the notion of the symmetric systems, introduced by the author in [34]. Symmetric systems play a fundamental role throughout this paper when we deal with the energy functional given in (1.16) and when we study system (1.15) with a general nonlinearity H(u).…”
Section: Introductionmentioning
confidence: 99%
“…We now establish a stability inequality for minimal solutions of system (P ) λ,γ . Note that for the case of local operators this inequality is established by the author and Cowan in [14] and in [22].…”
Section: Stability Inequalitiesmentioning
confidence: 85%
“…For the classification of solutions of above nonlocal equations on the entire space we refer interested readers to [12,16,24,30], and for the local equations to [19][20][21] and references therein. For the case of systems, as discussed in [14,22,33], set Q := {(λ, γ) : λ, γ > 0} and define…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…Note also that ∂ v H 1 = ∂ u H 2 < 0 and ∂ v H 1 ∂ u H 2 > 0. For a similar notion of stability, we refer interested readers to [1,22] for the Allen-Cahn system and to [9,12,13,20,21,31] for systems with general nonlinearities on bounded and unbounded domains.…”
Section: Introductionmentioning
confidence: 99%