## Advances in Differential Equations

- Adv. Differential Equations
- Volume 10, Number 2 (2005), 223-240.

### A weak maximum principle for the linearized operator of $m$-Laplace equations with applications to a nondegeneracy result

#### Abstract

We consider the Dirichlet problem for positive solutions of the equation $ -\Delta_m (u) = f(u) \; $ in a bounded, smooth domain $\, \Omega $, with $f$ positive and locally Lipschitz continuous. We prove a weak maximum principle in small domains for the linearized operator that we exploit to prove a weak maximum principle for the linearized operator. We then consider the case $f(s)=s^q$ and prove a nondegeneracy result in weighted Sobolev spaces.

#### Article information

**Source**

Adv. Differential Equations Volume 10, Number 2 (2005), 223-240.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867889

**Mathematical Reviews number (MathSciNet)**

MR2106131

**Zentralblatt MATH identifier**

1122.35020

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B50: Maximum principles

#### Citation

Sciunzi, Berardino. A weak maximum principle for the linearized operator of $m$-Laplace equations with applications to a nondegeneracy result. Adv. Differential Equations 10 (2005), no. 2, 223--240. https://projecteuclid.org/euclid.ade/1355867889.