Existence and Asymptotic Behavior of Solutions for a Quasilinear Schrödinger–Poisson System in ℝ³ with a General Nonlinearity

In this paper, we are concerned with the following quasilinear Schrödinger–Poisson system in ℝ³ \(\begin{cases}-\Delta u+u+\lambda\phi u=f(u), & \text{in}\ \mathbb{R^{3},} \\-\Delta \phi - \varepsilon^{4}\Delta_{4}\phi=\lambda u^{2}, & \text{in} \ \mathbb{R}^{3},\end{cases}\) $$\{\begin{array}{ll}-\mathrm{\Delta }u+u+\lambda \varphi u=f(u),& \text{in}{\mathbb{R}}^{\mathbb{3}}\mathbb{,}\\ -\mathrm{\Delta }\varphi -{\epsilon }^4{\mathrm{\Delta }}_4\varphi =\lambda u^2,& \text{in}{\mathbb{R}}^3,\end{array}$$ where λ and ε are positive parameters, Δ₄u = div(∣∇u∣²∇u), f ∈ C(ℝ, ℝ) is a general nonlinearity introduced by Berestycki and Lions [Arch. Rational Mech. Anal., 1983, 82(4): 313–345]. We obtain the existence of a nontrivial solution for small λ and fixed ε by using a monotonicity trick of Jeanjean and truncation method. Furthermore, the asymptotic behavior of these solutions is studied as ε and λ tend to zero respectively. We prove that they converge to a nontrivial solution of a classic Schrödinger–Poisson system and a Schrödinger equation associated with it respectively. One ground state solution is also obtained under a further growth hypothesis on f.