Liouville Results and Asymptotics of Solutions of a Quasilinear Elliptic Equation with Supercritical Source Gradient Term
Marie‐Françoise Bidaut‐Véron
Abstract
Abstract We consider the elliptic quasilinear equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>u</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> {-\Delta_{m}u=u^{p}\lvert\nabla u\rvert^{q}} in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {\mathbb{R}^{N}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>≥</m:mo> <m:mi>m</m:mi> </m:mrow> </m:math> {q\geq m} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {p>0} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>m</m:mi> <m:mo><</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {1<m<N} . Our main result is a Liouville-type property, namely, all the positive <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {C^{1}} solutions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {\mathbb{R}^{N}} are constant. We also give their asymptotic behaviour; all the solutions in an exterior domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo>∖</m:mo> <m:msub> <m:mi>B</m:mi> <m:msub> <m:mi>r</m:mi> <m:mn>0</m:mn> </m:msub> </m:msub> </m:mrow> </m:math> {\mathbb{R}^{N}\setminus B_{r_{0}}} are bounded. The solutions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>B</m:mi> <m:msub> <m:mi>r</m:mi> <m:mn>0</m:mn> </m:msub> </m:msub> <m:mo>∖</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {B_{r_{0}}\setminus\{0\}} can be extended as continuous functions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:msub> <m:mi>r</m:mi> <m:mn>0</m:mn> </m:msub> </m:msub> </m:math> {B_{r_{0}}} . The solutions in <m:math xmlns:m="http://www.w3.org/1998/Math/