Ground state sign-changing solutions for a class of quasilinear Schrödinger equations

Abstract

Abstract In this paper, we consider the following quasilinear Schrödinger equation: − Δ u + V ( x ) u + κ 2 Δ ( u 2 ) u = K ( x ) f ( u ) , x ∈ R N , -\Delta u+V\left(x)u+\frac{\kappa }{2}\Delta \left({u}^{2})u=K\left(x)f\left(u),\hspace{1.0em}x\in {{\mathbb{R}}}^{N}, where N ≥ 3 N\ge 3 , κ > 0 \kappa \gt 0 , f ∈ C ( R , R ) f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) , V ( x ) V\left(x) and K ( x ) K\left(x) are positive continuous potentials. Under given conditions, by changing variables and truncation argument, the energy of ground state solutions of the Nehari type is achieved. We also prove the existence of ground state sign-changing solutions for the aforementioned equation. Our results are the generalization work of M. B. Yang, C. A. Santos, and J. Z. Zhou, Least action nodal solution for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity, Commun. Contemp. Math. 21 (2019), no. 5, 1850026, https://doi.org/10.1142/S0219199718500268.