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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $\\Omega \\subseteq \\R^n$ be an open set, and let $1 \u003c p \u003c \\infty$.\nFor a function $u \\colon \\Omega \\to \\R$ whose [[gradient|0ABK9V9Y64]] $\\nabla u$ is regular enough, the *$p$-Laplacian* of $u$ is\n$$\\Delta_p u = \\operatorname{div} \\big( \\abs{\\nabla u}^{p-2} \\nabla u \\big),$$\nwhere $\\operatorname{div}$ is the [[divergence|0MWV6ZTK6S]].\nLet $u$ be in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}_{\\mathrm{loc}}(\\Omega)$.\nThe equation $\\Delta_p u = 0$ holds in the [[weak sense|SJZX81A783]] if\n$$\\int_\\Omega \\abs{\\nabla u}^{p-2} \\, \\nabla u \\cdot \\nabla \\varphi \\, dx = 0$$\nfor every smooth function $\\varphi$ with compact support in $\\Omega$.\nSuch a $u$ is called *$p$-harmonic*.\n\n## Remarks\n\nFor $p = 2$, $\\Delta_2 = \\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nExample: on $\\R^n \\setminus \\set{0}$ with $p \\ne n$, the function $u(x) = \\abs{x}^{(p-n)/(p-1)}$ is $p$-harmonic.\nIndeed, 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 $\\beta = (p - n)/(p - 1)$.\nNon-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 $x_1 \\ne 0$.\nThe weak form is the Euler–Lagrange equation of $\\int_\\Omega \\abs{\\nabla u}^p \\, dx$.\nSome authors call $-\\Delta_p$ the $p$-Laplacian.\n",
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