Ordinary p-Laplacian systems with nonlinear boundary conditions

This paper is concerned with the existence of solutions for the boundary value problem -(vertical bar u'vertical bar(p-2)u')' + epsilon vertical bar u vertical bar(p-2)u = del F(t,u), in (0,T), -((vertical bar u'vertical bar(p-2)u')(0),-(vertical bar u'vertical bar(p-2)u')(T)) is an element of partial derivative j(u(0), u(T)), where epsilon >=, 0, p is an element of (1, infinity) are fixed, j : R-N x R-N -> (-infinity, +infinity] is a proper, convex and lower semicontinuous function and F: (0, T) X R-N -> R is a Caratheodory mapping, continuously differentiable with respect to the second variable and satisfies some usual growth conditions. Our approach is a variational one and relies on Szulkin's critical point theory [A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincare Anal. Non Lineaire 3 (1986) 77-109]. We obtain the existence of solutions in a coercive case as well as the existence of nontrivial solutions when the corresponding Euler-Lagrange functional has a "mountain pass" geometry. (c) 2005 Elsevier Inc. All rights reserved.