On Schechter's linking theorems

The existence theorems of critical points due to M. Schechter [M. Schechter, The saddle point alternative, Amer. J. Math. 117 (1995) 1603–1626] are developed to the case of sign-changing critical points. A new relationship between the linking without value-splitting property and the sign-changing (weak) Palais–Smale sequence is established. The new abstract theorems are applied to find nodal solutions to elliptic equations under rather weak hypotheses.