On the behaviour of ∞-harmonic functions near isolated points Original Research Article

In this work we study the behaviour of ∞-harmonic functions near isolated exceptional points. We discuss two situations: (i) when points are in the interior and (ii) when points are on a flat portion of the boundary, on which a solution vanishes. We consider both bounded and unbounded solutions. Under the assumptions of boundedness, solutions are well-behaved in either case. We show that unbounded solutions in (i) change sign, and provide some lower bounds for growth rates valid for both (i) and (ii). In (ii), if the solution has one sign then we can also find an upper bound. We also look at the case of the ball.