Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities

Mingqi, Xiang; Rădulescu, Vicenţiu D.; Zhang, Binlin

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:

$$\left\{\begin{array}{cc}M({\left[\right(u,v\left)\right]}_{s,p}^p+{∥(u,v)∥}_{p,V}^p)({(-\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p-2}u)=\lambda H_u(x,u,v)+\frac{\alpha }{p_s^{*}}{\left|v\right|}^{\beta }{\left|u\right|}^{\alpha -2}u\hfill & \mathrm{in}{ℝ}^N\hfill \\ M({\left[\right(u,v\left)\right]}_{s,p}^p+{∥(u,v)∥}_{p,V}^p)({(-\Delta )}_p^{s}v+V\left(x\right){\left|u\right|}^{p-2}u)=\lambda H_v(x,u,v)+\frac{\beta }{p_s^{*}}{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v\hfill & \mathrm{in}{ℝ}^N\hfill \end{array}\right.$$

where {\left(–\Delta \right)}_p^s is the fractional p–Laplace operator with 0<s<1<p<N/s,\alpha ,\beta >1 with \alpha +\beta =p_s^{*},M:{ℝ}_0^{+}\to {ℝ}_0^{+} is a continuous function, V:{ℝ}^N\to {ℝ}^{+} is a continuous function, \lambda >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 {ℝ}^N.

Classification : 35D30, 35R11, 35A15, 47G20
Keywords: Integro–differential operator, Schrödinger–Kirhhoff system, critical nonlinearity, variational methods

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