In this paper, a fourth-order nonlinear difference equation is considered. By making use of the critical point theory, we establish various sets of sufficient conditions for the existence and nonexistence of solutions for Neumann boundary value problem and give some new results. Results obtained generalize and complement the existing ones.
We study a class of submanifolds, called Generalized Cauchy-Riemann (GCR) lightlike submanifolds of indefinite Sasakian manifolds
as an umbrella of invariant, screen real, contact CR lightlike subcases  and real hypersurfaces . We prove existence
and non-existence theorems and a characterization theorem on minimal GCR-lightlike submanifolds.
andnonexistence of positive entire solutions of second order semilinear elliptic systems, Funkcial. Ekvac. 42 (1999), 241-260. MR 2001i :35051
Existenceandnonexistence of positive entire solutions of second order