Ordinary p-Laplacian systems with nonlinear boundary conditions

This paper is concerned with the existence of solutions for the boundary value problem −(|u |p−2u ) +ε|u|p−2u=∇F(t,u), in (0,T ), ((|u |p−2u )(0),−(|u |p−2u )(T )) ∈ ∂j (u(0), u(T )), where ε 0, p ∈ (1,∞) are fixed, j :RN × RN → (−∞,+∞] is a proper, convex and lower semicontinuous function and F : (0,T ) ×RN →R is a Carathéodory 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. Poincaré Anal. Non Linéaire 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.  2005 Elsevier Inc. All rights reserved