| Volume 10, Number 5, 2020, Pages 2121-2144 DOI：10.11948/20190338 |
| Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity |
| Guofeng Che,Haibo Chen |
| Keywords:Kirchhoff equations, critical Sobolev exponent, Hartree-type nonlinearity, concentration-compactness principle. |
| Abstract: |
| This paper is concerned with the following Kirchhoff-type equations
\left\{
\begin{array}{ll}
\displaystyle
-\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u
+ V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array}
\right.
where f(x,u)=λK(x)|u|q−2u+Q(x)|u|4u, a>0, b, μ≥0 are constants, α∈(0,3), p∈[2,3), q∈[2p,6) and ε, λ>0 are parameters. Under some mild conditions on V(x), K(x) and Q(x), we prove that the above system possesses a ground state solution uε with exponential decay at infinity for λ>0 and ε small enough. Furthermore, uε concentrates around a global minimum point of V(x) as ε→0. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature. |
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