The semi-classical of multiple sign-changing solutions for a class of quasi-linear Schrödinger equations

Abstract This paper is concerned with studying the semi-classical problem for a class of quasi-linear Schrödinger equations: − ϵ 2 Δ u + V ( x ) u − ϵ 3 Δ ( u 2 ) u = | u | p − 2 u , x ∈ R N ( N ≥ 3 ) , $$-{{\epsilon}}^{2}{\Delta}u+V\left(x\right)u-{{\epsilon}}^{3}{\Delta}\left({u}^{2}\right)u=\vert u{\vert }^{p-2}u, x\in {\mathbb{R}}^{N}\left(N\ge 3\right),$$ where potential function V ( x ) has been pre-determined, ϵ > 0, p ∈ (2, 2*) with 2 * = 2 N N − 2 ${2}^{{\ast}}=\frac{2N}{N-2}$ . The problem is highly sensitive to the parameters ϵ 2 and ϵ 3 , which vanish at different rates. Using variational methods, we demonstrate that multiple sign-changing solutions with higher topological structures can be precisely localized.