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′)′+ɛ|u|p−2u=∇F(t,u),in (0,T),((|u′|p−2u′)(0),−(|u′|p−2u′)(T))∈∂j(u(0),u(T)), where ɛ⩾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.