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2008 Variational methods for ordinary $p$-Laplacian systems with potential boundary conditions
Petru Jebelean
Adv. Differential Equations 13(3-4): 273-322 (2008).


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.


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Petru Jebelean. "Variational methods for ordinary $p$-Laplacian systems with potential boundary conditions." Adv. Differential Equations 13 (3-4) 273 - 322, 2008.


Published: 2008
First available in Project Euclid: 18 December 2012

zbMATH: 1177.34021
MathSciNet: MR2482419

Primary: 34B15
Secondary: 34A60, 34L30, 47J30, 47N20, 49K24

Rights: Copyright © 2008 Khayyam Publishing, Inc.


Vol.13 • No. 3-4 • 2008
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