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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 $$u\not\in W_{1,p}$$.

##### MSC:
 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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##### References:
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