Construction of infinitely many solutions for two-component Bose-Einstein condensates with nonlocal critical interaction
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Cited by 2 publications
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Existence and classification of positive solutions for coupled purely critical Kirchhoff system
We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.
Existence and classification of positive solutions for coupled purely critical Kirchhoff system
We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.
Synchronized vector solutions for the nonlinear Hartree system with nonlocal interaction
We are concerned with the following nonlinear Hartree system − Δ u + P 1 ( | x | ) u = α 1 | x | − 1 ∗ u 2 u + β | x | − 1 ∗ v 2 u in R 3 , − Δ v + P 2 ( | x | ) v = α 2 | x | − 1 ∗ v 2 v + β | x | − 1 ∗ u 2 v in R 3 , $$\begin{cases}-{\Delta}u+{P}_{1}\left(\vert x\vert \right)u={\alpha }_{1}\left(\vert x{\vert }^{-1}\ast {u}^{2}\right)u+\beta \left(\vert x{\vert }^{-1}\ast {v}^{2}\right)u\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ -{\Delta}v+{P}_{2}\left(\vert x\vert \right)v={\alpha }_{2}\left(\vert x{\vert }^{-1}\ast {v}^{2}\right)v+\beta \left(\vert x{\vert }^{-1}\ast {u}^{2}\right)v\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \end{cases}$$ where P 1(r) and P 2(r) are positive radial potentials, α 1 > 0, α 2 > 0 and β ∈ R $\beta \in \mathbb{R}$ is a coupling constant. We first study nondegeneracy of ground states for the limit system of the above problem. As applications, we show that the nonlinear Hartree system has infinitely many non-radial positive synchronized solutions, whose energy can be arbitrarily large.
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