Multiplicity results for critical p $p$ -biharmonic problems

Abstract

We prove new multiplicity results for some critical growth $p$-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 $\lambda >0$. In particular, the number of solutions goes to infinity as $\lambda \to \infty$. We also give an explicit lower bound on $\lambda$ 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 $p=2$. The proofs are based on an abstract critical point theorem.