Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term

2020/08/24 by Marie-Françoise Bidaut-Veron, Bidaut-Veron, Marie-Françoise · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #Applied mathematics #Bounded function #Constant (computer programming) #Domain (mathematical analysis) #Elliptic curve #FOS: Mathematics #Geometric Analysis and Curvature Flows #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Nabla symbol #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Omega #Physics #Pure mathematics #Quantum mechanics #Supercritical fluid #Term (time) #Thermodynamics #Type (biology) #math.AP

paper · pdf · open access · doi:10.48550/arxiv.2008.10220

published in arXiv (Cornell University) (Cornell University)

arxiv created 2020/08/24 · openalex publication_date 2020/08/24 · arxiv updated 2020/08/25 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We consider the elliptic quasilinear equation --Δ m u = u p |∇u| q in R N with q ≥ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also give their asymptotic behaviour : all the solutions in an exterior domain R N \B r0 are bounded. The solutions in B r0 0 can be extended as a continuous functions in B r0. The solutions in R N 0 has a finite limit l ≥ 0 as |x| → ∞. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman's type estimate for the equation satisfied by the gradient.

Citations

• Classification of solutions of some nonlinear elliptic equations
• Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities
• Global and local behavior of positive solutions of nonlinear elliptic equations
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

Cited by

• Classification of positive solutions to a class of Laplace equations with a gradient term
• Liouville results for (p,q)-Laplacian elliptic equations with source terms involving gradient nonlinearities
• Classification of solutions of an elliptic Hamilton-Jacobi equation

Related

• A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms
• Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms
• Gradient Estimates for Solutions To Quasilinear Elliptic Equations with Critical Sobolev Growth and Hardy Potential
• Liouville results for (p,q)-Laplacian elliptic equations with source terms involving gradient nonlinearities
• Positive solutions of quasilinear elliptic equations with Fuchsian potentials in Wolff class