Extremal parameter for double phase problem with concave-convex nonlinearity

Note: Please see pdf for full abstract with equations. In this work, we study the following problem −Δ p u − div(μ(x)|∇u| q−2 ∇u) = λf(x)|u| γ−2 u + g(x)|u|r−2u in Ω, u = 0 on ∂Ω, where Ω ⊂ R N ,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 ∈ C c (Ω) ∩ L∞(Ω) and μ ≢ 0 . The objective of this work is to explore the optimal control on λ to apply Nehari manifold Mathematics Subject Classification (2010). Primary 35A02; Secondary 35A15, 35B32.