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
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https://hal.science/hal-03868763
Soumis le : mercredi 23 novembre 2022-23:22:59
Dernière modification le : vendredi 3 mai 2024-14:34:43
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Identifiants
- HAL Id : hal-03868763 , version 1
- ARXIV : 1907.06816
- DOI : 10.1515/ans-2019-2070
Citer
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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