On a class of critical double phase problems

Abstract

In this paper we study a class of double phase problems involving critical growth, namelywhere Ω ⊂ R N is a bounded Lipschitz domain, 1 < ϑ < p < q < N , q p < 1 + 1 N and µ(•) is a nonnegative Lipschitz continuous weight function. The operator involved is the so-called double phase operator, which reduces to the p-Laplacian or the (p, q)-Laplacian when µ ≡ 0 or inf µ > 0, respectively. Based on variational and topological tools such as truncation arguments and genus theory, we show the existence of λ * > 0 such that the problem above has infinitely many weak solutions with negative energy values for any λ ∈ (0, λ * ).