On finding sign-changing solutions

Some parameter-depending linking theorems are established, which allow to produce a bounded and sign-changing Palais–Smale sequence. For even functionals, a parameter-depending fountain theorem is obtained which provides infinitely many bounded and sign-changing Palais–Smale sequences. A variant mountain pass theorem is built in cones which yields bounded, positive and negative Palais–Smale sequences. The usual Palais–Smale type compactness condition and its variants are completely not necessary for these theories. More exact locations of the critical sequences can be determined. The abstract results are applied to the Schrödinger equation with (or without) critical Sobolev exponents:−Δu+V(x)u=β|u|2*−2u+f(x,u),x∈RN,β⩾0, where 2*2* is the critical Sobolev exponent. The existence of (multiple) sign-changing solutions is obtained. The positive and negative solutions are also gained as by-products. We will also show that this Schrödinger problem with jumping nonlinearity is independent of the Fučík spectrum.