Ground states of Kirchhoff equations via Pohozˇaev–Nehari manifold: existence,concentration and nonexistence.

This article is dedicated to the following Kirchhoff-type problem with Hartree-type nonlinearity: - (ε 2 a + ε b ∫ R 3 | ∇ u | 2 d x) ▵ u + V (x) u = ε μ - 3 K (x) (W μ (x) ∗ | u | p ) | u | p - 2 u , u ∈ H 1 (R 3) , <graphic href="43034_2022_245_Article_Equ53.gif"></graphic> where a , b ≥ 0 are constants, ε > 0 is a parameter, 0 < μ < 3 , p ∈ [ 2 , 6 - μ) , W μ (x) is convolution kernel, V(x) is nonnegative continuous external potential and K (x) ∈ C 1 (R 3) . Under some suitable assumptions on V(x) and K(x), we prove the existence and concentration of a positive ground state solution as ε → 0 via Poho z ˇ aev–Nehari manifold method. If K(x) is a positive constant, as its supplementary results, we also obtain the nonexistence of nontrivial solutions for p ≥ 6 - μ . [ABSTRACT FROM AUTHOR]