Existence of positive ground state solutions for Kirchhoff type problems

Abstract In this paper, we study the existence of positive ground state solutions for the nonlinear Kirchhoff type problem { − ( a + b ∫ R 3 | ∇ u | 2 ) △ u + V ( x ) u = f ( u ) in  R 3 , u ∈ H 1 ( R 3 ) , u > 0 in  R 3 , where a , b > 0 are constants, f ∈ C ( R , R ) is subcritical near infinity and superlinear near zero and satisfies the Berestycki–Lions condition. By using an abstract critical point theorem established by Jeanjean and a new global compactness lemma, we show that the above problem has at least a positive ground state solution. Our result generalizes the results of Li and Ye (2014) concerning the nonlinearity f ( u ) = | u | p − 1 u with p ∈ ( 2 , 5 ) .

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