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Article Dans Une Revue Advanced Nonlinear Studies Année : 2020

A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient

Résumé

Abstract We consider the elliptic equation - Δ ⁢ u = u q ⁢ | ∇ ⁡ u | p {-\Delta u=u^{q}|\nabla u|^{p}} in ℝ n {\mathbb{R}^{n}} for any p > 2 {p>2} and q > 0 {q>0} . We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in [2], where the case 0 < p < 2 {0

Dates et versions

hal-03868763 , version 1 (23-11-2022)

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Roberta Filippucci, Patrizia Pucci, Philippe Souplet. A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient. Advanced Nonlinear Studies, 2020, 20 (2), pp.245-251. ⟨10.1515/ans-2019-2070⟩. ⟨hal-03868763⟩
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