## 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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