## Advances in Differential Equations

- Adv. Differential Equations
- Volume 9, Number 9-10 (2004), 1185-1200.

### Antimaximum principle for quasilinear problems

David Arcoya and Julio D. Rossi

#### Abstract

In this paper, for a bounded, smooth domain $\Omega \subset {{\mathbb {R}}}^N$, $h\in L^r(\Omega)$, $r>N/2$, $g\in L^s(\partial \Omega)$, $s>N-1$, we prove the maximum and antimaximum principle for the quasilinear boundary-value problem $- \mbox{div} ( A(u) \nabla u) +u = h$ in $\Omega$, $ A(u) \nabla u \cdot \eta = \lambda f(u) + g$, on $\partial \Omega$, where $A$ is elliptic and bounded and $f$ is asymptotically linear. The sharpness of this result ($r>N/2$ and $s>N-1$) is discussed for the linear boundary-value problem, $-\Delta u + u = h$ in $\Omega$, $\partial u/\partial\eta = \lambda u + g $ on $\partial \Omega$.

#### Article information

**Source**

Adv. Differential Equations Volume 9, Number 9-10 (2004), 1185-1200.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867918

**Mathematical Reviews number (MathSciNet)**

MR2098070

**Zentralblatt MATH identifier**

1241.35059

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35B50: Maximum principles 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory

#### Citation

Arcoya, David; Rossi, Julio D. Antimaximum principle for quasilinear problems. Adv. Differential Equations 9 (2004), no. 9-10, 1185--1200. https://projecteuclid.org/euclid.ade/1355867918.