On the Kirchhoff problems involving critical Sobolev exponent

Abstract This paper deals with the following Kirchhoff type problem (0.1) − a + b ∫ R N | ∇ u | 2 Δ u = λ K ( x ) u q − 1 + u 2 ∗ − 1 , u > 0 , x ∈ R N , where N ≥ 3 , K ∈ L 1 ( R N ) ∩ L ∞ ( R N ) , constants a , b > 0 , the parameter λ > 0 and 2 ≤ q 2 ∗ ≔ 2 N ∕ ( N − 2 ) . Using the finite-dimensional reduction approach, we prove that problem (0.1) admits at least a positive solution.

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