Construction of singular solutions to the p-harmonic equation and its limit equation for \(p=\infty\)
Here, all solutions of the form \(u=r^ kf(\Phi)\) to the p-harmonic equation, \(div(| \nabla u|^{p-2}\nabla u)=0\) \((p>2)\) in the plane are determined. One main result is a representation formula for such solutions. Further, solutions with an isolated singularity at the origin are constructed (Theorem 1). Graphical illustrations are given at the end of the paper. Finally, all solutions \(u=r^ kf(\phi)\) of the limit equation for \(p=\infty\), \[ u^ 2_ xu_{xx}+2u_ xu_ yu_{xy}+u^ 2_ yu_{yy}=0, \] are constructed, some of which have a ''strong'' singularity at the origin (Theorem 2).
scientific article; zbMATH DE number 3955537
scientific article