Least energy sign-changing solution for degenerate Kirchhoff double phase problems
Ángel Crespo‐Blanco, Leszek Gasiński, Patrick Winkert
Abstract
In this paper we study the following nonlocal Dirichlet equation of double phase type−ψ[∫Ω(|∇u|pp+μ(x)|∇u|qq)dx]G(u)=f(x,u)in Ω,u=0on ∂Ω, where G is the double phase operator given byG(u)=div(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)u∈W01,H(Ω), Ω⊆RN, N≥2, is a bounded domain with Lipschitz boundary ∂Ω, 1<p<N, p<q<p⁎=NpN−p, 0≤μ(⋅)∈L∞(Ω), ψ(s)=a0+b0sϑ−1 for s∈R, with a0≥0, b0>0 and ϑ≥1, and f:Ω×R→R is a Carathéodory function that grows superlinearly and subcritically. We prove the existence of two constant sign solutions (one is positive, the other one negative) and of a sign-changing solution which turns out to be a least energy sign-changing solution of the problem above. Our proofs are based on variational tools in combination with the quantitative deformation lemma and the Poincaré-Miranda existence theorem.