On an elliptic Kirchhoff-type problem depending on two parameters

Let Ω ⊂ R be a bounded domain with smooth boundary, and let K : [0,+∞[→ R be a given continuous function. If n ≥ 2, we denote by A the class of all Carathéodory functions φ : Ω×R → R such that sup (x,t)∈Ω×R |φ(x, t)| 1 + |t|q < +∞ , where 0 < q < n+2 n−2 if n > 2 and 0 < q < +∞ if n = 2. While, when n = 1, we denote by A the class of all Carathéodory functions φ : Ω ×R → R such that, for each r > 0, the function x → sup|t|≤r |φ(x, t)| belongs to L (Ω). Given φ ∈ A, consider the following Kirchhoff-type problem