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- W3120481805 abstract "<p style='text-indent:20px;'>In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrödinger system <p style='text-indent:20px;'><disp-formula> <label>1</label> <tex-math id=E1> begin{document}$ begin{equation} left{begin{array}{ll} -iD_0Psi-(D_1D_1+D_2D_2)Psi+VPsi = |Psi|^{p-2}Psi, partial_0A_1-partial_1A_0 = -frac 12ilambda[overline{Psi}D_2Psi-Psioverline{D_2Psi}], partial_0A_2-partial_2A_0 = frac 12ilambda[overline{Psi}D_1Psi-Psioverline{D_1Psi}], partial_1A_2-partial_2A_1 = -frac12lambda|Psi|^2. end{array} right. end{equation} $end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>with an external potential <inline-formula><tex-math id=M1>begin{document}$ V(x) $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=M2>begin{document}$ D_{0} = partial_{t}+ilambda A_{0} $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=M3>begin{document}$ D_{k} = partial_{x_k}-ilambda A_{k}, , k = 1,2, $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=M4>begin{document}$ (x_1,x_2,t)in mathbb{R}^{2,1} $end{document}</tex-math></inline-formula> are covariant derivatives, <inline-formula><tex-math id=M5>begin{document}$ lambda $end{document}</tex-math></inline-formula> is the coupling number. Suppose that <inline-formula><tex-math id=M6>begin{document}$ V(x) $end{document}</tex-math></inline-formula> satisfies <inline-formula><tex-math id=M7>begin{document}$ lim_{|x|toinfty}V(x) = +infty $end{document}</tex-math></inline-formula>, we show for <inline-formula><tex-math id=M8>begin{document}$ 2<p<4 $end{document}</tex-math></inline-formula> that there exists <inline-formula><tex-math id=M9>begin{document}$ lambda^*>0 $end{document}</tex-math></inline-formula> such that if <inline-formula><tex-math id=M10>begin{document}$ 0<lambda<lambda^* $end{document}</tex-math></inline-formula>, problem (1) has two nontrivial static solutions <inline-formula><tex-math id=M11>begin{document}$ (Psi_lambda, A_0^lambda, A_1^lambda,A_2^lambda) $end{document}</tex-math></inline-formula>. Moreover, there also exists <inline-formula><tex-math id=M12>begin{document}$ tildelambda>0 $end{document}</tex-math></inline-formula> such that if <inline-formula><tex-math id=M13>begin{document}$ lambda>tildelambda $end{document}</tex-math></inline-formula>, problem (1) has no nontrivial solutions. While for <inline-formula><tex-math id=M14>begin{document}$ p>4 $end{document}</tex-math></inline-formula> we assume in addition that <inline-formula><tex-math id=M15>begin{document}$ xcdot nabla V(x)geq 0 $end{document}</tex-math></inline-formula>, then problem (1) admits a mountain pass solution for all <inline-formula><tex-math id=M16>begin{document}$ lambda>0 $end{document}</tex-math></inline-formula>." @default.
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- W3120481805 date "2021-01-01" @default.
- W3120481805 modified "2023-09-24" @default.
- W3120481805 title "Solutions to Chern-Simons-Schrödinger systems with external potential" @default.
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- W3120481805 doi "https://doi.org/10.3934/dcdss.2021008" @default.
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