  We prove an abstract critical point theorem based on a cohomological index
theory that produces pairs of nontrivial critical points with nontrivial higher
critical groups. This theorem yields pairs of nontrivial solutions that are
neither local minimizers nor of mountain pass type for problems with combined
nonlinearities. Applications are given to subcritical and critical
$p$-Laplacian problems, Kirchhoff type nonlocal problems, and critical
fractional $p$-Laplacian problems.

Analysis of PDEs