On sign-changing solutions for resonant (p,q)-Laplace equations

Abstract

. We provide two existence results for sign-changing solutions to the Dirichlet problem for the family of equations − ∆ p u − ∆ q u = α | u | p − 2 u + β | u | q − 2 u , where 1 < q < p and α , β are parameters. First, we show the existence in the resonant case α ∈ σ ( − ∆ p ) for sufficiently large β , thereby generalizing previously known results. The obtained solutions have negative energy. Second, we show the existence for any β > λ 1 ( q ) and sufficiently large α under an additional nonresonant assumption, where λ 1 ( q ) is the first eigenvalue of the q -Laplacian. The obtained solutions have positive energy.