p-harmonic function

Remarks

Every weak solution agrees almost everywhere with a continuous one, which is even of class $C^{1,\alpha}_{\mathrm{loc}}$ for some $\alpha > 0$, so the continuity requirement only picks a representative. For $p = 2$ the condition is $\int \nabla u \cdot \nabla \varphi = 0$, and by Weyl's lemma the $2$-harmonic functions are the classical harmonic functions. Example: for $p \ne n$, the function $u(x) = \abs{x}^{(p - n)/(p - 1)}$ is $p$-harmonic in $\R^n \setminus \set{0}$. Indeed, for a radial function $u(r)$ the equation reads $\big(r^{n-1} \abs{u'}^{p-2} u'\big)' = 0$, and here $\abs{u'}^{p-2} u'$ is a constant multiple of $r^{1-n}$. For $p = n$ the function $\log \abs{x}$ plays the same role. Non-example: $u(x) = x_1^2$ on $\R^n$ is not $p$-harmonic. Here $\abs{\nabla u}^{p-2} \nabla u = 2^{p-1} \abs{x_1}^{p-2} x_1\, e_1$, whose divergence $2^{p-1}(p - 1)\abs{x_1}^{p-2}$ is positive for $x_1 \ne 0$. Equivalently, $u$ is $p$-harmonic in $\Omega$ when it is continuous and minimizes $\int_D \abs{\nabla v}^p \, dx$ among all $v$ with $v - u \in W^{1,p}_0(D)$, for every open $D$ with compact closure in $\Omega$.