Ground State Solutions of Schrödinger-Kirchhoff Equations with Potentials Vanishing at Infinity

Abstract

In this paper, we deal with the following Schrödinger-Kirchhoff equation with potentials vanishing at infinity: − ε 2 a + ε b ∫ ℝ 3 ∇ u 2 Δ u + V x u = K x u p − 1 u in ℝ 3 and u > 0 , u ∈ H 1 ℝ 3 , where V x ~ x − α and K x ~ x − β with 0 < α < 2 , and β > 0 . We first prove the existence of positive ground state solutions u ε ∈ H 1 ℝ 3 under the assumption that σ < p < 5 for some σ = σ α , β , then we show that u ε concentrates at a global minimum point of A x = V 2 / p − 1 − 1 / 2 x / K 2 / p − 1 x .