Multiplicity results for critical p$p$‐biharmonic problems

Abstract

AbstractWe prove new multiplicity results for some critical growth ‐biharmonic problems in bounded domains. More specifically, we show that each of the problems considered here has arbitrarily many solutions for all sufficiently large values of a certain parameter . In particular, the number of solutions goes to infinity as . We also give an explicit lower bound on in order to have a given number of solutions. This lower bound will be in terms of an unbounded sequence of eigenvalues of a related eigenvalue problem. Our multiplicity results are new even in the semilinear case . The proofs are based on an abstract critical point theorem.