scholarly journals Asymptotic behavior of ground states of generalized pseudo-relativistic Hartree equation

2019 ◽  
pp. 1-27
Author(s):  
P. Belchior ◽  
H. Bueno ◽  
O.H. Miyagaki ◽  
G.A. Pereira
Keyword(s):  
Asymptotic Behavior ◽  
Ground States ◽  
Hartree Equation

scholarly journals On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qingxuan Wang ◽  
Binhua Feng ◽  
Yuan Li ◽  
Qihong Shi
Keyword(s):  
Ground State ◽  
Blow Up ◽  
Asymptotic Properties ◽  
Ground States ◽  
Threshold Value ◽  
Hartree Equation ◽  
Existence And Stability ◽  
Blow Up Rate

<p style='text-indent:20px;'>We consider the semi-relativistic Hartree equation with combined Hartree-type nonlinearities given by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_t \psi = \sqrt{-\triangle+m^2}\, \psi+\beta(\frac{1}{|x|^\alpha}\ast |\psi|^2)\psi-(\frac{1}{|x|}\ast |\psi|^2)\psi\ \ \ \text{on $\mathbb{R}^3$.} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\alpha&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \beta&gt;0 $\end{document}</tex-math></inline-formula>. Firstly we study the existence and stability of the maximal ground state <inline-formula><tex-math id="M3">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M4">\begin{document}$ N = N_c $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ N_c $\end{document}</tex-math></inline-formula> is a threshold value and can be regarded as "Chandrasekhar limiting mass". Secondly, we analyse blow-up behaviours of maximal ground states <inline-formula><tex-math id="M6">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \beta\rightarrow 0^+ $\end{document}</tex-math></inline-formula>, and the optimal blow-up rate with respect to <inline-formula><tex-math id="M8">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> will be calculated.</p>

Mass Concentration and Asymptotic Uniqueness of Ground State for 3-Component BEC with External Potential in ℝ2

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yuzhen Kong ◽  
Qingxuan Wang ◽  
Dun Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Ground State ◽  
Mass Concentration ◽  
Ground States ◽  
Unique Positive Solution ◽  
Energy Estimation ◽  
Concentration Phenomena ◽  
Existence And Nonexistence ◽  
Bose Einstein

Abstract We investigate the ground states of 3-component Bose–Einstein condensates with harmonic-like trapping potentials in ℝ 2 {\mathbb{R}^{2}} , where the intra-component interactions μ i {\mu_{i}} and the inter-component interactions β i ⁢ j = β j ⁢ i {\beta_{ij}=\beta_{ji}} ( i , j = 1 , 2 , 3 {i,j=1,2,3} , i ≠ j {i\neq j} ) are all attractive. We display the regions of μ i {\mu_{i}} and β i ⁢ j {\beta_{ij}} for the existence and nonexistence of the ground states, and give an elaborate analysis for the asymptotic behavior of the ground states as β i ⁢ j ↗ β i ⁢ j * := a ∗ + 1 2 ⁢ ( a ∗ - μ i ) ⁢ ( a ∗ - μ j ) {\beta_{ij}\nearrow\beta_{ij}^{*}:=a^{\ast}+\frac{1}{2}\sqrt{{(a^{\ast}-\mu_{i% })(a^{\ast}-\mu_{j})}}} , where 0 < μ i < a ∗ := ∥ w ∥ 2 2 {0<\mu_{i}<a^{\ast}:=\|w\|_{2}^{2}} are fixed and w is the unique positive solution of Δ ⁢ w - w + w 3 = 0 {\Delta w-w+w^{3}=0} in H 1 ⁢ ( ℝ 2 ) {H^{1}(\mathbb{R}^{2})} . The energy estimation as well as the mass concentration phenomena are studied, and when two of the intra-component interactions are equal, the nondegeneracy and the uniqueness of the ground states are proved.

scholarly journals Ground states for the pseudo-relativistic Hartree equation with external potential

2015 ◽  
Vol 145 (1) ◽  
pp. 73-90 ◽  
Author(s):  
Silvia Cingolani ◽  
Simone Secchi
Keyword(s):  
Ground State ◽  
Schrodinger Equation ◽  
Scalar Potential ◽  
Ground States ◽  
Hartree Equation ◽  
External Potential ◽  
Decay Estimates ◽  
Asymptotic Decay ◽  
Ground State Solutions ◽  
Symmetric Convolution

We prove the existence of positive ground state solutions to the pseudo-relativistic Schrödinger equationwhere N ≥ 3, m > 0, V is a bounded external scalar potential and W is a radially symmetric convolution potential satisfying suitable assumptions. We also provide some asymptotic decay estimates of the found solutions.

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