In this paper, we study the multiplicity and concentration of solutions for the following critical fractional Schrödinger–Poisson system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\u03f5}^{2s}{(-\u25b5)}^{s}u+V\left(x\right)u+\varphi u=f\left(u\right)+{\left|u\right|}^{{2}_{s}^{*}-2}u\hfill & \text{in}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathbb{R}}^{3},\hfill \\ {\u03f5}^{2t}{(-\u25b5)}^{t}\varphi ={u}^{2}\hfill & \text{in}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathbb{R}}^{3},\hfill \end{array}\right.\end{array}$$ |

Keywords: Fractional Schrödinger–Poisson system, positive solution, critical growth, variational method

^{1}; Zhang, Jianjun

^{2, 3}

@article{COCV_2017__23_4_1515_0, author = {Liu, Zhisu and Zhang, Jianjun}, title = {Multiplicity and concentration of positive solutions for the fractional {Schr\"odinger{\textendash}Poisson} systems with critical growth}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1515--1542}, publisher = {EDP-Sciences}, volume = {23}, number = {4}, year = {2017}, doi = {10.1051/cocv/2016063}, mrnumber = {3716931}, zbl = {06811887}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016063/} }

TY - JOUR AU - Liu, Zhisu AU - Zhang, Jianjun TI - Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1515 EP - 1542 VL - 23 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016063/ DO - 10.1051/cocv/2016063 LA - en ID - COCV_2017__23_4_1515_0 ER -

%0 Journal Article %A Liu, Zhisu %A Zhang, Jianjun %T Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1515-1542 %V 23 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016063/ %R 10.1051/cocv/2016063 %G en %F COCV_2017__23_4_1515_0

Liu, Zhisu; Zhang, Jianjun. Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 4, pp. 1515-1542. doi : 10.1051/cocv/2016063. http://archive.numdam.org/articles/10.1051/cocv/2016063/

Lévy processes-from probability theory to finance and quantum groups. Notices Am. Math. Soc. 51 (2004) 320–1331.

,C. Alves and O. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in ${R}^{N}$ via penalization method. To appear in Cal. Var. Partial Differ. Equ. (2016). | MR

On bound states concentrating on spheres for the Maxwell-Schrödinger equation. SIAM J. Math. Anal. 37 (2005) 321–342. | DOI | MR | Zbl

and ,On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27 (2010) 779–791. | DOI | Numdam | MR | Zbl

, and ,Ground state solutions for nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345 (2008) 90–108. | DOI | MR | Zbl

and ,Multiple bound states for the Schrödinger–Poisson problem. Commun. Contemp. Math. 10 (2008) 391–404. | DOI | MR | Zbl

and ,Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Cal. Var. Partial Differ. Equ. 2 (1994) 29–48. | DOI | MR | Zbl

and ,An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11 (1998) 283–293. | DOI | MR | Zbl

and ,Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82 (1983) 313–345. | DOI | MR | Zbl

and ,A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 32 (2015) 875–900. | DOI | Numdam | MR | Zbl

, , and ,A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 8 (1983) 486–490. | DOI | MR | Zbl

and ,Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26 (2013) 479–494. | DOI | MR | Zbl

and ,Concentration phenomenon for fractional nonlinear Schrödinger equations. Commun. Pure Appl. Anal. 13 (2014) 2359–2376. | DOI | MR | Zbl

and ,Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295 (2004) 225–236. | DOI | MR | Zbl

and ,Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Differ. Equ. 256 (2014) 858–892. | DOI | MR | Zbl

, and ,Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. Anal. Partial Differ. Equ. 8 (2015) 1165–1235. | MR | Zbl

, , and ,Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. | DOI | MR | Zbl

, and ,The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7 (1982) 77–116. | DOI | MR | Zbl

, and ,Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 142 (2012) 1237–1262. | DOI | MR | Zbl

, and ,On the variational principle. J. Math. Anal. Appl. 47 (1974) 324–353. | DOI | MR | Zbl

,Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity 28 (2015) 1937–1961. | DOI | MR | Zbl

, and ,Uniqueness and non degeneracy of ground states for ${(-\u25b3)}^{s}+Q-{Q}^{\alpha +1}=0$ in $R$. Acta Math. 210 (2013) 261–318. | Zbl

and ,Uniqueness of radial solutions for the fractional laplacian. Commun. Pure. Appl. Math. 69 (2016) 1671–1726. | DOI | MR | Zbl

, and ,A. Giammetta, Fractional Schrödinger–Poisson-Slater system in one dimension. Preprint (2014). | arXiv

Multiplicity and concentration of positive solutions for the Schrödinger–Poisson equations. Z. Angew. Math. Phys. 62 (2011) 869–889. | DOI | MR | Zbl

,Existence and concentration of ground states for Schrödinger–Poisson equations with critical growth. J. Math. Phys. 53 (2012) 023702. | DOI | MR | Zbl

and ,On concentration of positive bound states for the Schrödinger–Poisson problem with potentials. Adv. Nonlinear Stud. 8 (2008) 573–595. | DOI | MR | Zbl

and ,Fractional Schrödinger equation. Phys. Rev. E 66 (2002) 056108. | DOI | MR

,On ground state solutions for the Schrödinger–Poisson equations with critical growth. J. Math. Anal. Appl. 412 (2014) 435–448. | DOI | MR | Zbl

and ,Multiple semiclassical states for coupled Schrödinger–Poisson equations with critical exponential growth. J. Math. Phys. 56 (2015) 041505. | DOI | MR | Zbl

, and ,Improved Sobolev embeddings, profile decomposition, and concentration compactness for fractional Sobolev spaces. Calc. Var. 56 (2015) 041505. | MR | Zbl

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

,Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere. Math. Models Methods Appl. Sci. 15 (2005) 141–164. | DOI | MR | Zbl

,The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237 (2006) 655–674. | DOI | MR | Zbl

,On fractional Schrödinger equations in ${R}^{N}$ without the Ambrosetti-Rabinowitz condition. Topol. Methods Nonlinear Anal. 47 (2016) 19–41. | MR | Zbl

,Ground state solutions for nonlinear fractional Schrödinger equations in ${R}^{N}$. J. Math. Phys. 54 (2013) 031501. | DOI | MR | Zbl

,A Brezis-Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Anal. 6 (2013) 2445–2464. | DOI | MR | Zbl

and ,The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367 (2015) 67–102. | DOI | MR | Zbl

and ,Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55 (2006) 1155–1174. | DOI | MR | Zbl

,On fractional Schrödinger equations in ${R}^{N}$ with critical growth. J. Math. Phys. 54 (2013) 121502. | DOI | MR | Zbl

and ,Ground states for fractional Schrödinger equations with critical growth. Nonlinearity 27 (2014) 187–207. | DOI | MR | Zbl

and ,Concentrating solutions of nonlinear fractional Schrödinger equation with potentials. J. Differ. Equ. 258 (2015) 1106–1128. | DOI | MR | Zbl

and ,M. Struwe, Variational Methods; Application to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer-Verlag (2007). | MR

The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differ. Equ. 42 (2011) 21–41. | DOI | MR | Zbl

,Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in ${R}^{3}$. Calc. Var. Partial Differ. Equ. 48 (2013) 243–273. | DOI | MR | Zbl

, , and ,On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153 (1993) 229–244. | DOI | MR | Zbl

,On the existence of solutions for the Schrödinger–Poisson equations. J. Math. Anal. Appl. 346 (2008) 155–169. | DOI | MR | Zbl

and ,The existence and concentration of positive solutions for a nonlinear Schrödinger–Poisson system with critical growth. J. Math. Phys. 55 (2014) 031507. | DOI | MR | Zbl

,Fractional Schrödinger–Poisson systems with a general subcritical or critical nonlinearity. Adv. Nonlinear Stud. 16 (2016) 15–30. | DOI | MR | Zbl

, and ,*Cited by Sources: *