On a class of critical double phase problems

Farkas, Csaba and Fiscella, Alessio and Winkert, Patrick (2022) On a class of critical double phase problems. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. ISSN 0022-247X

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Abstract

In this paper we study a class of double phase problems involving critical growth, namely −div(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)=λ|u|ϑ−2u+|u|p∗−2u in Ω and u=0 on ∂Ω, where Ω⊂RN is a bounded Lipschitz domain, 1<ϑ<p<q<N, qp<1+1N 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 infμ>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∗).

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