Communications on Pure and Applied Analysis (CPAA)

Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions

Pages: 267 - 275, Volume 7, Issue 2, March 2008      doi:10.3934/cpaa.2008.7.267

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Petru Jebelean - West University of Timişoara, Department of Mathematics, Blvd. V. Pârvan no. 4, 300223 Timişoara, Romania (email)

Abstract: This paper deals with the existence of infinitely many solutions for the boundary value problem

$-( | u' | ^{p-2}u')' + \varepsilon |u|^{p-2}u= \nabla F(t,u), $ in $(0,T)$,

$((|u'|^{p-2}u')(0), $ $ -(|u'|^{p-2}u')(T))$ $\in \partial j(u(0), u(T)),$

where $\varepsilon \geq 0$, $p \in (1, \infty)$ are fixed, the convex function $j:\mathbb R^N \times \mathbb R^N \to (- \infty , +\infty ]$ is proper, even, lower semicontinuous and $F:(0,T) \times \mathbb R^N \to \mathbb R $ is a Carathéodory mapping, continuously differentiable and even with respect to the second variable.

Keywords:  Ordinary $p$-Laplacian system, critical point, Palais-Smale condition.
Mathematics Subject Classification:  Primary: 34B15, 34L30; Secondary: 49J40.

Received: July 2006;      Revised: April 2007;      Published: December 2007.