2023/04/12 by Luis J. Alı́as, Alias, Luis J., Giulio Colombo +3
Computer Science · Engineering · Mathematics · #35B53 #35J70 #35J92 #58J05 #5J62 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2304.05829
openalex publication_date 2023/04/12 · openalex created_date 2023/04/15 · openalex updated_date 2026/07/28
In this paper we prove some integral estimates on the minimal growth of the positive part u+ of subsolutions of quasilinear equations div A(x,u,∇ u) = V|u|p-2u on complete Riemannian manifolds M, in the non-trivial case u+\not≡ 0. Here A satisfies the structural assumption |A(x,u,∇ u)|p/(p-1) ≤ k ⟨ A(x,u,∇ u),∇ u⟩ for some constant k>0 and for p>1 the same exponent appearing on the RHS of the equation, and V is a continuous positive function, possibly decaying at a controlled rate at infinity. We underline that the equation may be degenerate and that our arguments do not require any geometric assumption on M beyond completeness of the metric. From these results we also deduce a Liouville-type theorem for sufficiently slowly growing solutions.