A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents
Authors: Zhen Maoding , Zhang Binlin , Han Xiumei 
Published: April 2021

<p style='text-indent:20px;'>In this paper, we study the following Kirchhoff-type fractional Schrödinger system with critical exponent in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} \left(a_{1}+b_{1}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^{s}u+u = \mu_1|u|^{2^*_s-2}u +\frac{\alpha\gamma}{2^*_s}|u|^{\alpha-2}u|v|^{\beta}+k|u|^{p-1}u,\\ \left(a_{2}+b_{2}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}v|^2dx\right)(-\Delta)^{s}v+v = \mu_2|v|^{2^*_s-2}v+ \frac{\beta\gamma}{2^*_s}|u|^{\alpha}|v|^{\beta-2}v+k|v|^{p-1}v,\\ \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ (-\Delta)^{s} $\end{document}</tex-math></inline-formula> is the fractional Laplacian, <inline-formula><tex-math id="M3">\begin{document}$ 0&lt;s&lt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N&gt;2s, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M5">\begin{document}$ 2_{s}^{\ast} = 2N/(N-2s) $\end{document}</tex-math></inline-formula> is the fractional critical Sobolev exponent, <inline-formula><tex-math id="M6">\begin{document}$ \mu_{1},\mu_{2},\gamma, k&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \alpha+\beta = 2_{s}^{\ast},\ 1&lt;p&lt;2_{s}^{\ast}-1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ a_{i},b_{i}\geq 0, $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ a_{i}+b_{i}&gt;0,\ \ i = 1,2 $\end{document}</tex-math></inline-formula>. By using appropriate transformation, we first get its equivalent system which may be easier to solve:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \begin{cases} (-\Delta)^{s}u+u = \mu_1|u|^{2^*_s-2}u+\frac{\alpha\gamma}{2^*_s}|u|^{\alpha-2}u|v|^{\beta}+k|u|^{p-1}u, \ \ x\in \mathbb{R}^N, \\ (-\Delta)^{s}v+v = \mu_2|v|^{2^*_s-2}v+\frac{\beta\gamma}{2^*_s}|u|^{\alpha}|v|^{\beta-2}v+k|v|^{p-1}v,\ \ x\in \mathbb{R}^N,\\ \lambda_{1}^{s}-a_{1}-b_{1}\lambda_{1}^{\frac{N-2s}{2}}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}u|^2dx = 0, \ \ \lambda_{1}\in \mathbb{R}^+,\\ \lambda_{2}^{s}-a_{2}-b_{2}\lambda_{2}^{\frac{N-2s}{2}}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}v|^2dx = 0, \ \ \lambda_{2}\in \mathbb{R}^+. \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Then, by using the mountain pass theorem, together with some classical arguments from Brézis and Nirenberg, we obtain the existence of solutions for the new system under suitable conditions. Finally, based on the equivalence of two systems, we get the existence of solutions for the original system. Our results give improvement and complement of some recent theorems in several directions.</p>