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Remarks on the strong maximum principle involving \(p\)-Laplacian. (English) Zbl 1362.35062

Summary: Let \(N\geq 1\), \(1<p<\infty\) and \(p^\ast=\max (1,p-1)\). Let \(\Omega\) be a bounded domain of \(\mathbf{R}^N\). We establish the strong maximum principle for the \(p\)-Laplace operator with a nonlinear potential term. More precisely, we show that every super-solution \(u \in W^{1, p^\ast}_{\mathrm{loc}}(\Omega)\) vanishes identically in \(\Omega\), if \(u\) is admissible and \(u = 0\) a.e on a set of positive \(p\)-capacity relative to \(\Omega\).

MSC:

35B50 Maximum principles in context of PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
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Full Text: Euclid