p-Laplacian

Remarks

For $p = 2$, $\Delta_2 = \Delta$ is the Laplacian. Example: on $\R^n \setminus \set{0}$ with $p \ne n$, the function $u(x) = \abs{x}^{(p-n)/(p-1)}$ is $p$-harmonic. Indeed, for $u = r^\beta$ with $r = \abs{x}$, the field $\abs{\nabla u}^{p-2} \nabla u$ is radial with size $\abs{\beta}^{p-1} r^{(\beta - 1)(p-1)}$, and its divergence vanishes when $r^{n-1} r^{(\beta-1)(p-1)}$ is constant, that is, when $\beta = (p - n)/(p - 1)$. Non-example: for $p = 3$, the function $u(x) = x_1^2$ is not $3$-harmonic, since $\abs{\nabla u} \nabla u = (4 x_1 \abs{x_1}, 0, \ldots, 0)$ has divergence $8 \abs{x_1} \ne 0$ for $x_1 \ne 0$. The weak form is the Euler–Lagrange equation of $\int_\Omega \abs{\nabla u}^p \, dx$. Some authors call $-\Delta_p$ the $p$-Laplacian.