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Nontrivial solutions for time fractional nonlinear Schrödinger-Kirchhoff type equations. (English) Zbl 1373.34015

From the summary and introduction: We study the existence of solutions via variational methods.
The objective of the present paper is to study time fractional Schrödinger-Kirchhoff type equation of the form \[ \begin{gathered} \Biggl(a+ b \int_{\mathbb{R}} |_{-\infty} D^\alpha_t u(t)|^2\,dt\Biggr)^{\theta-1}\, {_tD^\alpha_\infty}(_{-\infty} D^\alpha_t u(t))\\ +\mu V(t) u= f(t,),\quad t\in\mathbb{R},\;u\in H^\alpha(\mathbb{R}),\end{gathered} \] where \(\alpha\in (1/2,1]\), \({_\infty}D^\alpha_t\) and \({_t}D^\alpha_\infty\), respectively, denote left and right Liouville-Weyl fractional derivatives of order \(\alpha\) on \(\mathbb{R}\), \(a,b>0\) are constants, \(\mu>0\) is parameter \(\theta> 1\), \(f\in C(\mathbb{R}\times\mathbb{R}, \mathbb{R})\), and \(V:\mathbb{R}\to\mathbb{R}^+\) is a potential function.

MSC:

34A08 Fractional ordinary differential equations
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