Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions

Let Ω ⊂ R n \Omega \subset {{\bf{R}}}^{n} be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q ∈ ] 0 , 1 [ q\in ]0,1{[} , α ∈ L ∞ ( Ω ) \alpha \in {L}^{\infty }\left(\Omega ) , with α > 0 \alpha \gt 0 , and k ∈ N k\in {\bf{N}} . Then, the problem − tan ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x Δ u = α ( x ) u q in Ω u > 0 in