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:
$$\left\{\begin{array}{cc}M({\left[\right(u,v\left)\right]}_{s,p}^{p}+{\u2225(u,v)\u2225}_{p,V}^{p})({(-\Delta )}_{p}^{s}u+V\left(x\right){\left|u\right|}^{p-2}u)=\lambda {H}_{u}(x,u,v)+\frac{\alpha}{{p}_{s}^{*}}{\left|v\right|}^{\beta}{\left|u\right|}^{\alpha -2}u\phantom{\rule{0.166667em}{0ex}}\hfill & \mathrm{in}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1em}{0ex}}{\mathbb{R}}^{N}\hfill \\ M({\left[\right(u,v\left)\right]}_{s,p}^{p}+{\u2225(u,v)\u2225}_{p,V}^{p})({(-\Delta )}_{p}^{s}v+V\left(x\right){\left|u\right|}^{p-2}u)=\lambda {H}_{v}(x,u,v)+\frac{\beta}{{p}_{s}^{*}}{\left|u\right|}^{\alpha}{\left|v\right|}^{\beta -2}v\phantom{\rule{0.166667em}{0ex}}\hfill & \mathrm{in}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1em}{0ex}}{\mathbb{R}}^{N}\hfill \end{array}\right.$$ |
where \n {\left(\u2013\Delta \right)}_{p}^{s}\n is the fractional \n p\n–Laplace operator with \n 0<s<1<p<N/s,\alpha ,\beta >1\n with \n \alpha +\beta ={p}_{s}^{*},\phantom{\rule{0.166667em}{0ex}}M:{\mathbb{R}}_{0}^{+}\to {\mathbb{R}}_{0}^{+}\n is a continuous function, \n V\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}{\mathbb{R}}^{N}\to {\mathbb{R}}^{+}\n is a continuous function, \n \lambda >0\n 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 {\mathbb{R}}^{N}\n.
Keywords: Integro–differential operator, Schrödinger–Kirhhoff system, critical nonlinearity, variational methods
@article{COCV_2018__24_3_1249_0, 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://archive.numdam.org/articles/10.1051/cocv/2017036/} }
TY - JOUR AU - Mingqi, Xiang AU - Rădulescu, Vicenţiu D. AU - Zhang, Binlin TI - Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1249 EP - 1273 VL - 24 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017036/ DO - 10.1051/cocv/2017036 LA - en ID - COCV_2018__24_3_1249_0 ER -
%0 Journal Article %A Mingqi, Xiang %A Rădulescu, Vicenţiu D. %A Zhang, Binlin %T Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1249-1273 %V 24 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017036/ %R 10.1051/cocv/2017036 %G en %F COCV_2018__24_3_1249_0
Mingqi, Xiang; Rădulescu, Vicenţiu D.; Zhang, Binlin. Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 3, pp. 1249-1273. doi : 10.1051/cocv/2017036. http://archive.numdam.org/articles/10.1051/cocv/2017036/
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