## Advances in Differential Equations

- Adv. Differential Equations
- Volume 13, Number 3-4 (2008), 273-322.

### Variational methods for ordinary $p$-Laplacian systems with potential boundary conditions

#### Abstract

We present existence results for ordinary $p$-Laplacian systems of the form $$ -( | u' | ^{p-2}u')' = f(t,u), \; \; \hbox { in } [0,T], \tag*{(*)}$$ submitted to the general potential boundary condition $$ ((|u'|^{p-2}u')(0), -(|u'|^{p-2}u')(T) ) \in \partial j(u(0), u(T)).$$ Here, $p \in (1, \infty)$ is fixed, $j:\mathbb{R}^N \times \mathbb{R}^N \to (- \infty , +\infty ]$ is proper, convex and lower semicontinuous and $f:[0,T] \times \mathbb{R}^N \to \mathbb{R}^N $ is a Carathéodory mapping. Firstly, we deal with the potential case $f(t,u)=\nabla F(t,u)$, with $F:[0,T] \times \mathbb{R} ^N \to \mathbb{R} $ continuously differentiable with respect to the second variable. Secondly, the system will be a nonpotential one. Afterwards, instead of $(*)$ will be the differential inclusions system $$-( | u' | ^{p-2}u')' \in \overline {\partial } F(t,u), \; \; \hbox { in } [0,T], $$ where, this time, $F$ is only locally Lipschitz with respect to the second variable and $\overline {\partial } F(t,x)$ stands for Clarke's generalized gradient of $F(t, \cdot )$ at $x \in \mathbb{R}^N$. Several examples of applications are given.

#### Article information

**Source**

Adv. Differential Equations Volume 13, Number 3-4 (2008), 273-322.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867351

**Mathematical Reviews number (MathSciNet)**

MR2482419

**Zentralblatt MATH identifier**

1177.34021

**Subjects**

Primary: 34B15: Nonlinear boundary value problems

Secondary: 34A60: Differential inclusions [See also 49J21, 49K21] 34L30: Nonlinear ordinary differential operators 47J30: Variational methods [See also 58Exx] 47N20: Applications to differential and integral equations 49K24

#### Citation

Jebelean, Petru. Variational methods for ordinary $p$-Laplacian systems with potential boundary conditions. Adv. Differential Equations 13 (2008), no. 3-4, 273--322. https://projecteuclid.org/euclid.ade/1355867351