Concentration of solutions for a class of biharmonic and p-Laplacian equations

Abstract

This article is concerned with the existence of a ground state solution for the class of elliptic Kirchhoff–Boussinesq type problems Δ 2 u ± Δ p u + ( 1 + λV ( x ) ) u = f ( u ) + γ | u | 2 ∗ ∗ − 2 u in R N , where 2 < p < 2 ∗ = 2 N N − 2 for N ≥ 3 and 2 ∗ ∗ = ∞ for N = 3, N = 4, 2 ∗ ∗ = 2 N N − 4 for N ≥ 5 . Here f is a continuous function and the term 1 + λV ( x ) is the steep potential well introduced by Bartsch and Wang in [Bartsch T, Wang ZQ. Existence and multiplicity results for superlinear elliptic problems on R N . Commun Partial Differ Equ 1995;20:1725–1741]. The function f has subcritical growth and behaves like | u | q − 2 u with p < q < 2 ∗ ∗ . We show the existence of a ground state solution using variational methods considering the subcritical case, i.e. γ = 0 and the critical case, i.e. γ = 1 .