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arXiv · 2008.10220

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

Marie-Françoise Bidaut-Veron
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Abstract

We consider the elliptic quasilinear equation --$\Delta$ m u = u p |$\nabla$u| q in R N with q $\ge$ 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 $\ge$ 0 as |x| $\rightarrow$ $\infty$. 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.

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Marie-Françoise Bidaut-Veron. 2020-08-24. Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term. https://arxiv.org/abs/2008.10220

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