Positive solutions of a Kirchhoff–Schrödinger--Newton system with critical nonlocal term - Ying Zhou; Jun Lei; Yue Wang; Zonghong Xiong (Electronic Journal of Qualitative Theory of Differential Equations, 2022)
This paper deals with the following Kirchhoff–Schrödinger–Newton system with critical growth \begin{equation*} \begin{cases} \displaystyle-M\left(\int_{\Omega}|\nabla u|^2dx\right)\Delta u=\phi |u|^{2^*-3}u+\lambda|u|^{p-2}u, &\rm \mathrm{in}\ \ \Omega, \\ \displaystyle-\Delta \phi=|u|^{2^*-1}, &\rm \mathrm{in}\ \ \Omega, \\ \displaystyle u=\phi=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^N(N\geq3)$ is a smooth bounded domain, $M(t)=1+bt^{\theta-1}$ with $t>0$, $1<\theta<\frac{N+2}{N-2}$, $b>0$, $1<p<2$, $\lambda>0$ is a parameter, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent. By using the variational method and the Brézis–Lieb lemma, the existence and multiplicity of positive solutions are established.