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 theorem given in [20] that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.

Entire solutions of quasilinear symmetric systems

“…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].…”

“…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).…”

Regularity of extremal solutions of nonlocal elliptic systems

“…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].…”

“…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…”

Stable solutions of symmetric systems on Riemannian manifolds

“…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.…”