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Keywords:
fractional-Schrödinger-Poisson; quasi-linear term; perturbation method; variational method
fractional-Schrödinger-Poisson; quasi-linear term; perturbation method; variational method
Summary:
The existence of nontrivial solutions is considered for the fractional Schrödinger-Poisson system with double quasi-linear terms: $$ \begin{cases} (-\Delta )^{s}u+V(x)u+\phi u -{1\over 2}u (-\Delta )^{s}u^{2}=f(x,u), & x\in \mathbb {R}^{3} ,\\ (-\Delta )^{t} \phi = u^{2}, & x\in \mathbb {R}^{3}, \end{cases} $$ where $(-\Delta )^{\alpha }$ is the fractional Laplacian for $\alpha =s$, $t\in (0,1]$ 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.
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