Entire solutions of quasilinear symmetric systems

Abstract

We study the following quasilinear elliptic system for allSeveral celebrated operators such as the prescribed mean curvature, the Laplacian and the p-Laplacian operators fit in the above form, for appropriate Φ. We establish a Hamiltonian identity of the following form for all xn ∈ R R n−1 m i=1 2010 Mathematics Subject Classification. 35J45, 35J93, 35J92, 35J50.

Counterexamples to Modica's gradient estimate for systems arising in multi-phase transitions

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We construct a plethora of one-dimensional periodic solutions for a class of semilinear elliptic systems of phase transition type which violate Modica's gradient estimate. This complements a recent technical counterexample in [14] and is partly motivated by a recent open problem in [9].

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“…In fact, several properties related to (3) (with exponent n − 1) have recently been shown to hold in [4] for this class of systems in the case of energy minimal solutions (in the sense of Morse). In this regard, it is of interest to know whether the analog of Modica's gradient estimate (2) holds for bounded, entire solutions to this class of systems (see the related comments in [3] and Open Problem 1 in [9]). It is worth noting that a class of such systems which satisfy this property has been provided in [10].…”

Monotonicity and rigidity of solutions to some elliptic systems with uniform limits

“…with multi-well potential has attracted increasing attention in the last years. Beyond the aforementioned results regarding problem (1.3), we refer in particular to [15,27], where the authors proved, under some assumptions on the potential W , symmetry for monotone or stable solutions in dimension N = 2 or N = 3, and to [4], where the authors investigated rigidity properties of minimal solutions to suitable symmetric systems; see also [14,15,16,26,28] for related results in low dimension, regarding more general operators and possibly unbounded solutions. Notice that system (P ) falls within the general case (1.8), with…”

Stable solutions of symmetric systems on Riemannian manifolds

“…the notion of stability is given in [20][21][22] and references therein. We also refer interested readers to [4,5,18,38] for the following two-component elliptic system, originated in the Bose-Einstein condensation and nonlinear optics,…”

“…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.…”