标题： 一类涉及一般临界增长的Kirchhoff型问题的多个正解 显示英文标题

标题： Multiple positive solutions for a class of Kirchhoff type problems involving general critical growth

摘要： 在本文中，我们研究以下涉及临界增长的非线性 Kirchhoff 问题： $$ \left\{% \begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=|u|^4u+\lambda|u|^{q-2}u, u=0\ \ \text{on}\ \ \partial\Omega, \end{array}% \right. $$ 其中 $1<q<2$， $\lambda,\ a,\ b>0$是参数且 $\Omega$是 $\R^3$中的一个有界区域。 我们证明存在 $\lambda_1=\lambda_1(q,\Omega)>0$ 使得对于任意的 $\lambda\in(0,\lambda_1)$ 和 $a,\ b>0$，上述 Kirchhoff 问题至少有两个正解，其中一个为正的基态解。 我们还建立了当参数 $b\searrow 0$时基态解的收敛性性质。 更一般地，我们得到关于以下Kirchhoff问题的相同结果：$$ \left\{% \begin{array}{ll} -(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+u=Q(x)|u|^4u+{\lambda}f(x)|u|^{q-2}u, u\in H^1(\mathbb{R}^3), \end{array}% \right. $$对于任何$a,\ b>0$和$\lambda\in \big(0,\lambda_0(q,Q,f)\big)$在$f(x)$和$Q(x)$的某些条件之下。 最后，我们研究$\lambda_0$和$b$之间的依赖关系，以证明对于任何（大）$\lambda>0$，存在一个$b_0(\lambda)>0$，使得当$b>b_0(\lambda)$和$a>0$时上述结果成立。 显示英文摘要

摘要： In this paper, we study the following nonlinear Kirchhoff problem involving critical growth: $$ \left\{% \begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=|u|^4u+\lambda|u|^{q-2}u, u=0\ \ \text{on}\ \ \partial\Omega, \end{array}% \right. $$ where $1<q<2$, $\lambda,\ a,\ b>0$ are parameters and $\Omega$ is a bounded domain in $\R^3$. We prove that there exists $\lambda_1=\lambda_1(q,\Omega)>0$ such that for any $\lambda\in(0,\lambda_1)$ and $a,\ b>0$, the above Kirchhoff problem possesses at least two positive solutions and one of them is a positive ground state solution. We also establish the convergence property of the ground state solution as the parameter $b\searrow 0$. More generally, we obtain the same results about the following Kirchhoff problem: $$ \left\{% \begin{array}{ll} -(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+u=Q(x)|u|^4u+{\lambda}f(x)|u|^{q-2}u, u\in H^1(\mathbb{R}^3), \end{array}% \right. $$ for any $a,\ b>0$ and $\lambda\in \big(0,\lambda_0(q,Q,f)\big)$ under certain conditions of $f(x)$ and $Q(x)$. Finally, we investigate the depending relationship between $\lambda_0$ and $b$ to show that for any (large) $\lambda>0$, there exists a $b_0(\lambda)>0$ such that the above results hold when $b>b_0(\lambda)$ and $a>0$.