On the existence of nontrivial solutions for modified fractional Schrödinger-Poisson systems via perturbation method

Abstract

The existence of nontrivial solutions is considered for the fractional Schrödinger-Poisson system with double quasi-linear terms:

$$\begin{cases}(-\Delta)^{s}u+V(x)u+\phi u -{1\over2}u (-\Delta)^{s}u^{2}=f(x,u), & x\in\mathbb{R}^{3} , \\ (-\Delta)^{t} \phi= u^{2}, & x\in\mathbb{R}^{3},\end{cases}$$

where (−Δ)^α is the fractional Laplacian for α = s, t ∈ (0, 1] with s < t and 2t + 4s > 3. Under assumptions on V and f, we prove the existence of positive solutions and negative solutions for the above system by using perturbation method and the mountain pass theorem.