Extremal parameter for double phase problem with concave–convex nonlinearity

Abstract

In this work, we study the following problem −Δpu−div(μ(x)|∇u|q−2∇u)=λf(x)|u|γ−2u+g(x)|u|r−2uinΩ,u=0on∂Ω,where Ω⊂RN,N≥max{2,p} is a bounded smooth domain, 1<γ<p<q<r<p∗=Np/(N−p) and λ is a positive real parameter. The weights f,g:Ω→R are continuous and bounded functions, where g may change sign on Ω. The function 0≤μ∈Cc(Ω) and μ⁄≡0. The objective of this work is to explore the optimal control on λ to apply Nehari manifold idea of constrained minimization in order to establish the existence and multiplicity of solutions.