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

Marie-Françoise Bidaut-Veron (1)

1 IDP - Institut Denis Poisson

We consider the elliptic quasilinear equation −∆ 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.

Mots clés: Liouville property; Bernstein method; Keller-Osserman estimates

Domaines: Equations aux dérivées partielles [math.AP]

Marie-Françoise Bidaut-Veron. Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term. Advanced Nonlinear Studies, 2020, 21 (1), pp.57-76. ⟨10.1515/ans-2020-2109⟩. ⟨hal-02919420⟩