The \(p\)-harmonic system with measure-valued right hand side. (English) Zbl 0879.35052

For \(2-1/n<p<n\) one proves existence of a distributional solution \(u\) of the \(p\)-harmonic system \[ -\text{div}(|\nabla u|^{p-2}\nabla u)= \mu\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega, \] where \(\Omega\) is an open subset of \(\mathbb{R}^n\) (bounded or unbounded), \(u:\Omega\to\mathbb{R}^n\), and \(\mu\) is an \(\mathbb{R}^m\)-valued Radon measure of finite mass. For the solution \(u\) one establishes the Lorentz space estimate \[ |Du|_{Lq,\infty}+|u|_{Lq^*,\infty}\leq C|\mu|M_{1/(p-1)}, \] \(q=n/(n-1)(p-1)\) and \(q^*=n/(n-p)(p-1)\). The main step of the proof is to show that for suitable approximations the gradients \(Du_k\) converge. This is achieved by the choice of regularized test functions and a localization argument to compensate the fact that in general \(u\not\in W_{1,p}\).


35J60 Nonlinear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35D05 Existence of generalized solutions of PDE (MSC2000)
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