Multiple Normalized Solutions for Choquard equations involving Kirchhoff type perturbation

Zeng Liu


In this paper we study the existence of critical points of the $C^1$ functional \begin{equation*} E(u)=\frac{a}{2}\int_{\bbrn}|\nabla u|^2dx+\frac{b}{4}\bigg(\int_{\bbrn}|\nabla u|^2dx\bigg)^2-\frac{1}{2p}\int_{\bbrn}(I_\alpha*|u|^p)|u|^pdx \end{equation*} under the constraint \begin{equation*} S_c=\bigg\{u\in H^1(\bbrn)\ \bigg\vert\ \int_{\bbrn}|u|^2dx=c^2\bigg\}, \end{equation*} where $a> 0$, $b> 0$, $N\geq3$, $\alpha\in(0,N)$, $ (N+\alpha)/{N}< p< (N+\alpha)/$ $(N-2)$ and $I_{\alpha}$ is the Riesz Potential. When $p$ belongs to different ranges, we obtain the threshold values separating the existence and nonexistence of critical points of $E$ on $S_c$. We also study the behaviors of the Lagrange multipliers and the energies corresponding to the constrained critical points when $c\to 0$ and $c\to +\infty$, respectively.


Choquard equation; Kirchhoff equation; mountain pass geometry; normalized solutions

Full Text:



C. Alves, F. Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations 263 (2017), 3943–3988.

J. Bellazzini, L. Jeanjean and T. Luo, Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc. 107 (2013), 303–339.

J. Bellazzini and G. Siciliano, Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal. 261 (2011), 2486–2507.

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math. 59 (2006), 330–343.

Y. Deng, L. Lu and W. Shuai, Constraint minimizers of mass critical hartree energy functionals: Existence and mass concentration, J. Math. Phys. 56 (2015), 249–261.

Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3 , J. Funct. Anal. 269 (2015), 3500–3527.

G. Figueiredo, N. Ikoma and J. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal. 213 (2014), 931–979.

Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in R3 involving critical Sobolev exponents, Calc. Var. Partial Differential Equations 54 (2015), 3067–3106.

X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 , J. Differential Equations 252 (2012), 1813–1834.

L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (1997), 1633–1659.

L. Jeanjean, T. Luo and Z. Wang, Multiple normalized solutions for quasi-linear Schrodinger equations, J. Differential Equations 259 (2016), 3894–3928.

G. Kirchhoff, Mechanik, Leipzig, Teubner, 1883.

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3 , J. Differential Equations 257 (2014), 566–600.

G. Li and H. Ye, The existence of positive solutions with prescribed L2 -norm for nonlinear Choquard equations, J. Math. Phys. 55 (2014), 251–268.

S. Li, J. Xiang and X. Zeng, Ground states of nonlinear Choquard equations with multiwell potentials, J. Math. Phys. 57 (2016), 081515.

Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012), 2285–2294.

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93–105.

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), 455–467.

I. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schröinger–Newton equations, Classical Quantum Gravity 15 (1998), 2733–2742.

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153–184.

V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), 6557–6579.

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations 52 (2015), 199–235.

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2014), 1550005.

V. Moroz and J. Van Schaftingen, A guide to Choquard equation, J. Fixed Point Theory Appl. 19 (2017), 773–813.

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Berlin, Akademie Verlag, 1954.

J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012), 2314–2351.

M. Yang, J. Zhang and Y. Zhang, Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity, Comm. Pure Appl. Anal. 16 (2017), 493–512.

H. Ye, Mass minimizers and concentration for nonlinear Choquard equations in RN , Topol. Methods Nonlinear Anal. 48 (2016), 393–417.

H. Ye, The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations, Math. Methods Appl. Sci. 38 (2015), 2663–2679.

H. Ye, The existence of normalized solutions for L2 -critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys. 66 (2015), 1483–1497.

X. Zeng and Y. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Appl. Math. Lett. 74 (2017), 52–59.


  • There are currently no refbacks.

Partnerzy platformy czasopism