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In this paper, we study the following Kirchhoff type problem − ( a + b ∫ R3 K(x)|∇u|2dx ) div(K(x)∇u) = λK(x)|x|β|u|q−2u + K(x)|u|4u, x ∈ R3, where K(x) = exp(|x|α/4) with α ≥ 2, β = (α − 2)(6 − q)/4 and the parameters a, b, λ > 0. When 6− 4 α < q < 6, we obtain a positive ground state solution for any λ > 0. When 2 < q < 4, we obtain a positive solution for λ > 0 small enough. In the proof we use variational methods.

Positive solutions for a Kirchhoff type problem with fast increasing weight and critical nonlinearity