标题： 具有临界不定非线性的分数Kirchhoff问题 显示英文标题

标题： Fractional Kirchhoff problem with critical indefinite nonlinearity

摘要： 我们研究了一类具有临界非线性的分数Kirchhoff方程正解的存在性和多重性，其形式为 \begin{equation*} M\left(\int_\Omega|(-\Delta)^{\frac{\alpha}{2}}u|^2dx\right)(-\Delta)^{\alpha} u= \lambda f(x)|u|^{q-2}u+|u|^{2^*_\alpha-2}u\;\; \text{in}\; \Omega,\;u=0\;\textrm{in}\;\mathbb R^n\setminus \Omega, \end{equation*} 其中 $\Omega\subset \mathbb R^n$是一个光滑有界区域， $ M(t)=a+\varepsilon t, \; a, \; \varepsilon>0,\; 0<\alpha<1, \; 2\alpha<n<4\alpha$和 $ \; 1<q<2$。 此处 $2^*_\alpha={2n}/{(n-2\alpha)}$是分数临界Sobolev指数， $\lambda$是一个正参数，系数 $f(x)$是一个允许变号的实值连续函数。 通过基于Nehari流形技巧思想的变分方法，我们将次线性和超线性项的效果结合起来，证明了我们的主要结果。 显示英文摘要

摘要： We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*} M\left(\int_\Omega|(-\Delta)^{\frac{\alpha}{2}}u|^2dx\right)(-\Delta)^{\alpha} u= \lambda f(x)|u|^{q-2}u+|u|^{2^*_\alpha-2}u\;\; \text{in}\; \Omega,\;u=0\;\textrm{in}\;\mathbb R^n\setminus \Omega, \end{equation*} where $\Omega\subset \mathbb R^n$ is a smooth bounded domain, $ M(t)=a+\varepsilon t, \; a, \; \varepsilon>0,\; 0<\alpha<1, \; 2\alpha<n<4\alpha$ and $ \; 1<q<2$. Here $2^*_\alpha={2n}/{(n-2\alpha)}$ is the fractional critical Sobolev exponent, $\lambda$ is a positive parameter and the coefficient $f(x)$ is a real valued continuous function which is allowed to change sign. By using a variational approach based on the idea of Nehari manifold technique, we combine effects of a sublinear and a superlinear term to prove our main results.