Abstract
AbstractThis paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the Hörmander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that generate the Heisenberg group, giving explicit conditions on the sign and size of the first and zero-th order terms in the equation. The optimality of the conditions is shown via several examples. A sequel of this paper applies the methods to other Carnot groups and to Grushin geometries.
Funder
Università degli Studi di Padova
Publisher
Springer Science and Business Media LLC
Reference41 articles.
1. Adamowicz, T., Kijowski, A., Pinamonti, A., Warhurst, B.: Variational approach to the asymptotic mean-value property for the $$p$$-Laplacian on Carnot groups. Nonlinear Anal. 198, 111893 (2020). 22
2. Adamowicz, T., Warhurst, B.: Three-spheres theorems for subelliptic quasilinear equations in Carnot groups of Heisenberg-type. Proc. Am. Math. Soc. 144(10), 4291–4302 (2016)
3. Armstrong, S.N., Sirakov, B.: Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10(3), 711–728 (2011)
4. Armstrong, S.N., Sirakov, B., Smart, C.K.: Fundamental solutions of homogeneous fully nonlinear elliptic equations. Commun. Pure Appl. Math. 64(6), 737–777 (2011)
5. Bardi, M., Cesaroni, A.: Liouville properties and critical value of fully nonlinear elliptic operators. J. Differ. Equ. 261(7), 3775–3799 (2016)
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