scholarly journals Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kazuhiro Kurata ◽  
Yuki Osada
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Ground State ◽  
State Energy ◽  
Wave Interaction ◽  
Vector Solution ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Positive Threshold ◽  
Schrödinger System

<p style='text-indent:20px;'>In this paper, we consider the asymptotic behavior of the ground state and its energy for the nonlinear Schrödinger system with three wave interaction on the parameter <inline-formula><tex-math id="M1">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M2">\begin{document}$ \gamma \to \infty $\end{document}</tex-math></inline-formula>. In addition we prove the existence of the positive threshold <inline-formula><tex-math id="M3">\begin{document}$ \gamma^* $\end{document}</tex-math></inline-formula> such that the ground state is a scalar solution for <inline-formula><tex-math id="M4">\begin{document}$ 0 \le \gamma &lt; \gamma^* $\end{document}</tex-math></inline-formula> and is a vector solution for <inline-formula><tex-math id="M5">\begin{document}$ \gamma &gt; \gamma^* $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Ground State Solution for an Autonomous Nonlinear Schrödinger System

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Min Liu ◽  
Jiu Liu
Keyword(s):  
Ground State ◽  
Critical Sobolev Exponent ◽  
Growth Conditions ◽  
Ground State Solution ◽  
State Solution ◽  
Subcritical Growth ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Sobolev Exponent ◽  
Schrödinger System

In this paper, we study the following autonomous nonlinear Schrödinger system (discussed in the paper), where λ , μ , and ν are positive parameters; 2 ∗ = 2 N / N − 2 is the critical Sobolev exponent; and f satisfies general subcritical growth conditions. With the help of the Pohožaev manifold, a ground state solution is obtained.

scholarly journals Synchronized and ground-state solutions to a coupled Schrödinger system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mohammad Ali Husaini ◽  
Chuangye Liu
Keyword(s):  
Bounded Domain ◽  
Ground State ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
Coupled Schrödinger System ◽  
Nonlinear Schrödinger System ◽  
Schrödinger System

<p style='text-indent:20px;'>In this paper, we study the following coupled nonlinear Schrödinger system of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{array}{l} -\Delta u_i-\kappa_iu_i = g_i(u_i)+\lambda\partial_iF(\vec{u}), \\ \vec{u} = (u_1,u_2,\cdots,u_m), u_i\in D_0^{1,2}(\Omega), \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for <inline-formula><tex-math id="M1">\begin{document}$ m = 2,3 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> is a bounded domain or <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ F(t_1,t_2\cdots,t_m)\in C^1(\mathbb{R}^m,\mathbb{R}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \kappa_i\in\mathbb{R} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ g_i\in C(\mathbb{R}) \ (i = 1,2,\cdots,m) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is large enough. In this work we mainly focus on the existence of fully nontrivial ground-state solutions and synchronized ground-state solutions under certain conditions.</p>

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