Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1249-1273.

In this paper, we investigatve the existence of solutions for critical Schrödinger–Kirchhoff type systems drien by nonlocal integro–differential operators. As a particular case, we consider the following system:

M ( [ ( u , v ) ] s , p p + ( u , v ) p , V p ) ( ( - Δ ) p s u + V ( x ) u p - 2 u ) = λ H u ( x , u , v ) + α p s * v β u α - 2 u in N M ( [ ( u , v ) ] s , p p + ( u , v ) p , V p ) ( ( - Δ ) p s v + V ( x ) u p - 2 u ) = λ H v ( x , u , v ) + β p s * u α v β - 2 v in N

where ( Δ ) p s is the fractional p –Laplace operator with 0 < s < 1 < p < N / s , α , β > 1 with α + β = p s * , M : 0 + 0 + is a continuous function, V : N + is a continuous function, λ > 0 is a real parameter. By applying the mountain pass theorem and Ekeland’s variational principle, we obtain the existence and asymptotic behaviour of solutions for the above systems under some suitable assumptions. A distinguished feature of this paper is that the above systems are degenerate, that is, the Kirchhoff function could vanish at zero. To the best of our knowledge, this is the first time to exploit the existence of solutions for fractional Schrödinger–Kirchhoff systems involving critical nonlinearities in N .

DOI : 10.1051/cocv/2017036
Classification : 35D30, 35R11, 35A15, 47G20
Mots clés : Integro–differential operator, Schrödinger–Kirhhoff system, critical nonlinearity, variational methods
Mingqi, Xiang 1 ; Rădulescu, Vicenţiu D. 1 ; Zhang, Binlin 1

1
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     author = {Mingqi, Xiang and R\u{a}dulescu, Vicen\c{t}iu D. and Zhang, Binlin},
     title = {Combined effects for fractional {Schr\"odinger{\textendash}Kirchhoff} systems with critical nonlinearities},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1249--1273},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {3},
     year = {2018},
     doi = {10.1051/cocv/2017036},
     zbl = {1453.35184},
     mrnumber = {3877201},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2017036/}
}
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Mingqi, Xiang; Rădulescu, Vicenţiu D.; Zhang, Binlin. Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1249-1273. doi : 10.1051/cocv/2017036. http://www.numdam.org/articles/10.1051/cocv/2017036/

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