The paper deals with the study of a class of semicoercive variational inequalities, as well as with some applications in the theory of von Kármán plates. In the first part of the paper there are proved several abstract results related to semicoerciveness in Banach spaces. The proofs are essentially based on some classical results in nonlinear functional analysis, such as: Garding’s inequality, Peetre’s lemma, and Poincaré’s inequality. The main result in this part of the paper is an existence theorem for variational inequalities involving ker \(A\)-invariant nonlinearities for a semicoercive nonlinear operator \(A\). This abstract result is then illustrated with an application to a friction problem for thin plates in the framework of von Kármán nonlinear elasticity.