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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be an open set and let $1 \u003c p \u003c \\infty$.\nLet $u$ lie in the local [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}_{\\mathrm{loc}}(\\Omega)$.\nIt is a *weak solution* of the [[p-Laplace equation|XJMYGY0ZZF]] $\\operatorname{div}(\\abs{\\nabla u}^{p-2} \\nabla u) = 0$ in $\\Omega$ if\n$$\\int_\\Omega \\abs{\\nabla u}^{p-2}\\, \\nabla u \\cdot \\nabla \\varphi \\, dx = 0 \\quad \\text{for every } \\varphi \\in C_c^\\infty(\\Omega),$$\nwhere $\\nabla u$ is the [[weak gradient|C3EC148VZD]].\nA *$p$-harmonic function* in $\\Omega$ is a continuous weak solution of this equation.\n\n## Remarks\n\nEvery weak solution agrees almost everywhere with a continuous one, which is even of class $C^{1,\\alpha}_{\\mathrm{loc}}$ for some $\\alpha \u003e 0$, so the continuity requirement only picks a representative.\nFor $p = 2$ the condition is $\\int \\nabla u \\cdot \\nabla \\varphi = 0$, and by Weyl's lemma the $2$-harmonic functions are the classical [[harmonic functions|0DEB0JMN0W]].\nExample: for $p \\ne n$, the function $u(x) = \\abs{x}^{(p - n)/(p - 1)}$ is $p$-harmonic in $\\R^n \\setminus \\set{0}$.\nIndeed, 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 $r^{1-n}$.\nFor $p = n$ the function $\\log \\abs{x}$ plays the same role.\nNon-example: $u(x) = x_1^2$ on $\\R^n$ is not $p$-harmonic.\nHere $\\abs{\\nabla u}^{p-2} \\nabla u = 2^{p-1} \\abs{x_1}^{p-2} x_1\\, e_1$, whose divergence $2^{p-1}(p - 1)\\abs{x_1}^{p-2}$ is positive for $x_1 \\ne 0$.\nEquivalently, $u$ 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 $\\Omega$.\n",
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