A constructive approach to positive solutions of Δ p u + f(u,∇u)≤0 on Riemannian manifolds

Wang, Yuzhao; Xiao, Jie

doi:10.1016/j.anihpc.2015.06.003

A constructive approach to positive solutions of Δ_p u + f(u,∇u)≤0 on Riemannian manifolds

Wang, Yuzhao ; Xiao, Jie

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1497-1507.

Résumé

Grigor'yan–Sun in [6] (with $p = 2$ ) and Sun in [10] (with $p > 1$ ) proved that if

\sup_{r ≫ 1} vol (B (x_{0}, r)) r^{\frac{p σ}{p - σ - 1}} {(\ln r)}^{\frac{p - 1}{p - σ - 1}} < \infty

then the only non-negative weak solution of

Δ_{p} u + u^{σ} \leq 0

on a complete Riemannian manifold is identically 0; moreover, the powers of r and

\ln r

are sharp. In this note, we present a constructive approach to the sharpness, which is flexible enough to treat the sharpness for

Δ_{p} u + f (u, \nabla u) \leq 0

. Our construction is based on a perturbation of the fundamental solution to the p-Laplace equation, and we believe that the ideas introduced here are applicable to other nonlinear differential inequalities on manifolds.

Zbl

DOI : 10.1016/j.anihpc.2015.06.003

Classification : 35J70, 58J05
Mots clés : Non-negative solution, Volume growth consideration, Complete Riemannian manifold

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     author = {Wang, Yuzhao and Xiao, Jie},
     title = {A constructive approach to positive solutions of {\ensuremath{\Delta}\protect\textsubscript{}            \protect\emph{p}         }         \protect\emph{u}         +         \protect\emph{f}(\protect\emph{u},\ensuremath{\nabla}\protect\emph{u})\ensuremath{\leq}0 on {Riemannian} manifolds},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1497--1507},
     publisher = {Elsevier},
     volume = {33},
     number = {6},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.06.003},
     zbl = {1353.58009},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2015.06.003/}
}

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Wang, Yuzhao; Xiao, Jie. A constructive approach to positive solutions of Δ_p         u         +         f(u,∇u)≤0 on Riemannian manifolds. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1497-1507. doi : 10.1016/j.anihpc.2015.06.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.06.003/

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