Fractional Schrödinger–Poisson–Kirchhoff type systems involving critical nonlinearities

Abstract

The paper is concerned with existence, multiplicity and asymptotic behavior of nonnegative solutions for a fractional Schrödinger–Poisson–Kirchhoff type system. As a consequence, the results can be applied to the special case (a+b‖u‖2)[(−Δ)su+V(x)u]+ϕk(x)|u|p−2u=λh(x)|u|q−2u+|u|2s∗−2uinR3,(−Δ)tϕ=k(x)|u|pinR3,‖u‖=∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy+∫R3V(x)|u|2dx1∕2,where a,b≥0 are two numbers, with a+b>0, 1<p<2s,t∗=3+2t3−2s, λ>0 is a parameter, s,t∈(0,1), (−Δ)s is the fractional Laplacian, k∈L63+2t−p(3−2s)(R3) may change sign, V:R3→[0,∞) is a potential function, 2s∗=6∕(3−2s) is the critical Sobolev exponent, 1<q<2s∗ and h∈L2s∗2s∗−q(R3). First, when θ<p<2s∗∕2, 2p≤q<2s∗ and λ is large enough, existence of nonnegative solutions is obtained by the mountain pass theorem. Moreover, we obtain that limλ→∞‖uλ‖=0. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when θ<p<2s∗∕2, 1<q<2 and λ is small enough, and we obtain that limλ→0‖uλ‖=0. Finally, we consider the system with double critical exponents, that is, p=2s,t∗, and obtain two nontrivial and nonnegative solutions in which one is least energy solution and another is mountain pass solution. The paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero. Hence the results of the paper are new even for the standard stationary Schrödinger–Poisson–Kirchhoff system.