Mass minimizers and concentration for nonlinear Choquard equations in ${\mathbb R}^N$

Hongyu Ye

DOI: http://dx.doi.org/10.12775/TMNA.2016.066

Abstract


In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$ E(u)=\frac{1}{2}\int_{\mathbb R^N}|\nabla u|^2+\frac{1}{2}\ds\int_{\mathbb R^N}V(x)|u|^2-\frac{1}{2p}\ds\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p $$% on $\widetilde{S}(c)=\{u\in H^1(\mathbb R^N)\mid \int_{\mathbb R^N}V(x)|u|^2< +\infty,\ |u|_2=c,\ c> 0\}$, where $N\geq1$, $\alpha\in(0,N)$, $({N+\alpha})/{N}\leq p< ({N+\alpha})/{(N-2)_+}$ and $I_\alpha\colon \mathbb R^N\rightarrow\mathbb R$ is the Riesz potential. We present sharp existence results for $E(u)$ constrained on $\widetilde{S}(c)$ when $V(x)\equiv0$ for all $({N+\alpha})/{N}\leq p< ({N+\alpha})/{(N-2)_+}$. For the mass critical case $p=({N+\alpha+2})/{N}$, we show that if $0\leq V\in L_{\rom{loc}}^{\infty}(\mathbb R^N)$ and $\lim\limits_{|x|\rightarrow+\infty}V(x)=+\infty$, then mass minimizers exist only if $0< c< c_*=|Q|_2$ and concentrate at the flattest minimum of $V$ as $c$ approaches $c_*$ from below, where $Q$ is a groundstate solution of $-\Delta u+u=(I_\alpha*|u|^{({N+\alpha+2})/{N}})|u|^{({N+\alpha+2})/{N}-2}u$ in $\mathbb R^N$.

Keywords


Choquard equation; mass concentration; normalized solutions; Sharp existence

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References


R.A. Adams and J.J. Fournier, Sobolev spaces, 2nd ed., Academic Press (2003).

J. Bellazzini, R.L. Frank and N. Visciglia, Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems, Math. Ann. 360 (2014), no. 3–4, 653- -673, DOI:10.1007/s00208-014-1046-2.MR3273640.

D.M. Cao and Q. Guo, Divergent solutions to the 5D Hartree equations, Colloq. Math. 125 (2) (2011), 255–287.

S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 140 (5) (2010), 973–1009.

P. Choquard, J. Stubbe and M. Vuffray, Stationary solutions of the Schrödinger–Newton model – an ODE approach, Differential Integral Equations 21 (2008), no. 7–8, 665–679.

Y.B. Deng, Y.J. Guo and L. Lu, On the collapse and concentration of Bose–Einstein condensates with inhomogeneous attractive interactions, Calc. Var. Partial Differential Equations, DOI: 10.1007/s00526-014-0779-9.

H. Genev and G. Venkov, Soliton and blow-up solutions to the time-dependent Schrödinger–Hartree equation, Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 5, 903–923.

Y.J. Guo and R. Seiringer, On the mass concentration for Bose–Einstein condensates with attractive interactions, Lett. Math. Phys. 104 (2014), 141–156.

Y.J. Guo, X.Y. Zeng and H.S. Zhou, Energy estimates and symmetry breaking in attractive Bose–Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire (2015), http://dx.doi.org/10.1016/j.anihpc.2015.01.005.

Y.J. Guo, Z.Q. Wang, X.Y. Zeng and H.S. Zhou, Properties of ground states of attractive Gross–Pitaevskiı̆ equations with multi-well potentials, arXiv: 1502.01839v1.

J. Krieger, E. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for L2 -critical Hartree NLS, Ann. H. Poincaré 10 (6) (2009), 1159–1205.

G.B. Li and H.-Y. Ye, The existence of positive solutions with prescribed L2 -norm for nonlinear Choquard equations, J. Math. Phys. 55 (2014), 121501; DOI: 10.1063/1.4902386.

M. Li and Z. Lin, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), no. 2, 455–467.

E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57 (1977), no. 2, 93–105.

E.H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. Math. 118 (2) (1983), no. 2, 349–374.

E.H. Lieb and B. Simon, The Hartree–Fock theory for Coulomb systems, Commun. Math. Phys. 53 (1977), 185–194.


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