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

This article is concerned with the existence of a ground state solution for the class of elliptic Kirchhoff–Boussinesq type problems Δ2u±Δpu+(1+λV(x))u=f(u)+γ|u|2∗∗−2uinRN, where 2<p<2∗=2NN−2 for N≥3 and 2∗∗=∞ for N = 3, N = 4, 2∗∗=2NN−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 RN. Commun Partial Differ Equ 1995;20:1725–1741. The function f has subcritical growth and behaves like |u|q−2u 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.