Semiclassical ground state solutions for a Choquard type equation in 2 with critical exponential growth
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 177-209.

In this paper we study a nonlocal singularly perturbed Choquard type equation

-ε 2 Δu+V(x)u= μ-2 1 |x| μ *P ( x ) G ( u )P(x)g(u)
in 2 , where ε is a positive parameter, 1 |x| μ with 0<μ<2 is the Riesz potential, * is the convolution operator, V(x), P(x) are two continuous real functions and G(s) is the primitive function of g(s). Suppose that the nonlinearity g is of critical exponential growth in 2 in the sense of the Trudinger-Moser inequality, we establish some existence and concentration results of the semiclassical solutions of the Choquard type equation in the whole plane. As a particular case, the concentration appears at the maximum point set of P(x) if V(x) is a constant.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017007
Classification : 35J25, 35J20, 35J60
Mots clés : Choquard equation, semiclassical solutions, Trudinger-Moser inequality, critical exponential growth
Yang, Minbo 1

1 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, P.R. China.
@article{COCV_2018__24_1_177_0,
     author = {Yang, Minbo},
     title = {Semiclassical ground state solutions for a {Choquard} type equation in $\mathbb{R}^{2}$ with critical exponential growth},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {177--209},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {1},
     year = {2018},
     doi = {10.1051/cocv/2017007},
     mrnumber = {3764139},
     zbl = {1400.35086},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2017007/}
}
TY  - JOUR
AU  - Yang, Minbo
TI  - Semiclassical ground state solutions for a Choquard type equation in $\mathbb{R}^{2}$ with critical exponential growth
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2018
SP  - 177
EP  - 209
VL  - 24
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2017007/
DO  - 10.1051/cocv/2017007
LA  - en
ID  - COCV_2018__24_1_177_0
ER  - 
%0 Journal Article
%A Yang, Minbo
%T Semiclassical ground state solutions for a Choquard type equation in $\mathbb{R}^{2}$ with critical exponential growth
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2018
%P 177-209
%V 24
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2017007/
%R 10.1051/cocv/2017007
%G en
%F COCV_2018__24_1_177_0
Yang, Minbo. Semiclassical ground state solutions for a Choquard type equation in $\mathbb{R}^{2}$ with critical exponential growth. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 177-209. doi : 10.1051/cocv/2017007. http://archive.numdam.org/articles/10.1051/cocv/2017007/

Adimurthi and S.L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of 2 involving critical exponent. Ann. Scuola. Norm. Sup. Pisa 17 (1990) 481–504. | Numdam | MR | Zbl

Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in R N and its applications. Int. Math. Res. Not. 13 (2010) 2394–2426. | MR | Zbl

C.O. Alves and G.M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in N . J. Differ. Equ. 246 (2009) 1288–1311. | DOI | MR | Zbl

C.O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in 2 . J. Differ. Equ. 261 (2016) 1933–1972. | DOI | MR | Zbl

C.O. Alves and M. Yang, Multiplicity and concentration behavior of solutions for a quasilinear Choquard equation via penalization method, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016) 23–58. | DOI | MR | Zbl

C.O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation. J. Differ. Equ. 257 (2014) 4133–4164. | DOI | MR | Zbl

C.O. Alves and M. Yang, Existence of solutions for a nonlinear Choquard equation in 2 with exponential critical growth. Preprint . | arXiv

A. Ambrosetti and A. Malchiodi, Concentration phenomena for nonlinear Schödinger equations: recent results and new perspectives, Perspectives in nonlinear partial differential equations. edited by H. Berestycki, M. Bertsch, F. E. Browder, L. Nirenberg, L.A. Peletier and L. Véron. Vol. 446 of Contemp. Math., Amer. Math. Soc. Providence, RI (2007) 19–30. | MR | Zbl

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schödinger equations with potentials. Arch. Ration. Mech. Anal. 159 (2001) 253–271. | DOI | MR | Zbl

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schröinger equations with a general nonlinearity. Arch. Rational Mech. Anal. 185 (2007) 185–200. | DOI | MR | Zbl

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486–490. | DOI | MR | Zbl

D. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in 2 . Commun. Partial Differ. Equ. 17 (1992) 407–435. | DOI | MR | Zbl

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schödinger equations with competing potential functions. J. Differ. Equ. 160 (2000) 118–138. | DOI | MR | Zbl

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

Y.H. Ding and X.Y. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math. 140 (2013) 51–82. | DOI | MR | Zbl

D.G. De Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in 2 with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3 (1995) 139–153. | DOI | MR | Zbl

J.M Do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two. Calc. Var. Partial Differ.l Equ. 38 (2010) 275–315. | DOI | MR | Zbl

A. Floer and A. Weinstein, Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential. J. Funct. Anal. 69 (1986) 397–408. | DOI | MR | Zbl

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ. 21 (2004) 287–318. | DOI | MR | Zbl

E. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics. AMS, Providence, Rhode island (2001). | Zbl

E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Studies Appl. Math. 57 (1976/77) 93–105. | DOI | MR | Zbl

P. L. Lions, The Choquard equation and related questions. Nonlinear Anal. TMA 4 (1980) 1063–1073. | DOI | MR | Zbl

Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in N . Indiana Univ. Math. J. 57 (2008) 451–480. | DOI | MR | Zbl

N. Lam and G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in N . J. Funct. Anal. 262 (2012) 1132–1165. | DOI | MR | Zbl

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE 2 (2009) 1–27. | DOI | MR | Zbl

G. Li, Some properties of weak solutions of nonlinear scalar field equations. Ann. Acad. Sci. Fenincae, Series A 14 (1989) 27–36. | MR | Zbl

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

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

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation. Calc. Var. Partial Differ. Equ. 52 (2015) 199–235. | DOI | MR | Zbl

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle. Akademie Verlag, Berlin (1954). | Zbl

M. Del Pino and P. Felmer, Local Mountain Pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4 (1996) 121–137. | DOI | MR | Zbl

P. Rabinowitz, On a class of nonlinear Schrödinger equations. Z. Ang. Math. Phys. 43 (1992) 270–291. | DOI | MR | Zbl

J. Wei and M. Winter, Strongly Interacting Bumps for the Schrödinger-Newton Equations. J. Math. Phys. 50 (2009) 012905. | DOI | MR | Zbl

X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions. SIAM J. Math. Anal. 28 (1997) 633–655. | DOI | MR | Zbl

M. Willem, Minimax Theorems, Birkhäuser (1996). | MR | Zbl

M. Yang, J. Zhang and Y. Zhang, Multi-peak solution for nonlinear Choquard equation with a general nonlinearity. Commun. Pure Appl. Anal. 16 (2017) 493–512. | DOI | MR | Zbl

Cité par Sources :