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

Petru Jebelean

#### 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

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