## Hiroshima Mathematical Journal

### Remarks on the strong maximum principle involving $p$-Laplacian

#### Abstract

Let $N\ge 1, 1 \lt p \lt \infty$ and $p^*=\max(1,p-1)$. 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 $u \in \Omega^{1, p^*}_{\mathrm{loc}}(\Omega)$ vanishes identically in $\Omega$, if $u$ is admissible and $u = 0$ a.e on a set of positive $p$-capacity relative to $\Omega$.

#### Article information

Source
Hiroshima Math. J., Volume 46, Number 3 (2016), 311-331.

Dates
Revised: 20 May 2016
First available in Project Euclid: 25 February 2017

https://projecteuclid.org/euclid.hmj/1487991624

Digital Object Identifier
doi:10.32917/hmj/1487991624

Mathematical Reviews number (MathSciNet)
MR3614300

Zentralblatt MATH identifier
1362.35062

Subjects
Primary: 35B50: Maximum principles
Secondary: 35J92: Quasilinear elliptic equations with p-Laplacian

#### Citation

Liu, Xiaojing; Horiuchi, Toshio. Remarks on the strong maximum principle involving $p$-Laplacian. Hiroshima Math. J. 46 (2016), no. 3, 311--331. doi:10.32917/hmj/1487991624. https://projecteuclid.org/euclid.hmj/1487991624