## Advances in Differential Equations

- Adv. Differential Equations
- Volume 8, Number 3 (2003), 337-356.

### A homotopy along $p$ for systems with a vector $p$-Laplace operator

M. García-Huidobro, R. Manásevich, and J. R. Ward

#### Abstract

We extend to the vector $p-$Laplace operator $(\phi_p(u'))'=(|u'|^{p-2}u')',$ $p>1,$ ($|\cdot|$ denotes the Euclidean norm in $\mathbb R^N$) a method that uses a suitable homotopy along $p$ to evaluate at the level $p=2$ a Leray-Schauder degree for an associated operator depending on $p.$ We apply this result to prove existence of nontrivial solutions to the system $$(\phi_p(u'))'=f(t,u)\quad\text{a.e.}\quad \mbox{in}\quad (0,T),\quad u(0)=0,\quad u(T)=0, $$ where $f:[0,T]\times\mathbb R^N\mapsto \mathbb R^N$ is a Carathéodory function, with $f(t,0)=0.$

#### Article information

**Source**

Adv. Differential Equations Volume 8, Number 3 (2003), 337-356.

**Dates**

First available in Project Euclid: 19 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355926857

**Mathematical Reviews number (MathSciNet)**

MR1948049

**Zentralblatt MATH identifier**

1039.34010

**Subjects**

Primary: 34B15: Nonlinear boundary value problems

Secondary: 47H11: Degree theory [See also 55M25, 58C30] 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05]

#### Citation

García-Huidobro, M.; Manásevich, R.; Ward, J. R. A homotopy along $p$ for systems with a vector $p$-Laplace operator. Adv. Differential Equations 8 (2003), no. 3, 337--356. https://projecteuclid.org/euclid.ade/1355926857