Existence of a Solution for a Class of Kirchhoff-Boussinesq Type Equations with Potential Changing Signs

This paper is concerned with the existence of a ground state solution for the following class of elliptic Kirchhoff-Boussinesq type problems given by Δ ² u ± Δ ₚ u + V(x)u= f(u) +γ |u|^2**-2u \ \ in \ \ RN, Δ 2 u ± Δ p u + V ( x ) u = f ( u ) + γ | u | 2 ∗ ∗ − 2 u in R N , where 2< p≤ 2*= 2N/N-2 2 < p ≤ 2 ∗ = 2 N N − 2 for N≥ 5 N ≥ 5 and 2**= 2N/N-4 2 ∗ ∗ = 2 N N − 4 . The potential V V can change sign and f f is a continuous function with subcritical growth and behaves like |u|q-2u | u | q − 2 u with p< q< 2** p < q < 2 ∗ ∗ . We show the existence of a ground state solution using variational methods, Splitting Lemma and considering the subcritical case, i.e, γ =0 γ = 0 and the critical case, i.e, γ =1 γ = 1 .