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A uniqueness result for very weak solutions of \(p\)-harmonic type equations. (English) Zbl 0854.35047
A generalization of the \(p\)-Harmonic equation \(\text{div}(|\nabla u|^{p- 2} \nabla u)= 0\); \(p> 1\), namely \[ \text{div}(a(x)|\nabla u|^{p- 2} \nabla u)= 0\tag{1} \] is considered, where \(a(x)\) belongs to the Muckenhoupt class \(A_p\). The author considers very weak solutions of (1), i.e. one assumes \(\int_{\mathbb{R}^n} a(x) |\nabla u|^{p- 2} \nabla u \nabla \varphi= 0\) for any \(\varphi\in {\mathcal C}^\infty_0(\mathbb{R}^n)\), where \(u\) belongs to the weighted Sobolev space \(W^{1, s}(\mathbb{R}^n, a)\) with \(\max\{1, p- 1\}< s< p\) and one proves that, if \(s\) is sufficiently close to \(p\), then \(u\equiv 0\).

MSC:
35J70 Degenerate elliptic equations
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