学术报告报告题目：Improvements and generalizations of Clark's theorem and applications
报 告 人：刘兆理 教授(首都师范大学)
报告时间：2019年11月23日（周六）上午10:00-11:00
报告地点：37号楼3A02室
邀 请 人：李用声 教授
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数学学院
2019年11月22日
报告摘要：
In critical point theory, Clark's theorem states as follows. Let $X$ be a Banach space, $\Phi\in C^1(X,\mathbb R)$. Assume $\Phi$ satisfies the (PS) condition, is even and bounded from below, and $\Phi(0) = 0$. If for any $k\in\mathbb N$, there exists a $k$-dimensional subspace $X^k$ of $X$ and $\rho_k>0$ such that $\sup_{X^k\cap S_{\rho_k}}\Phi<0$, where $S_\rho=\{u\in X\, |\, \|u\|=\rho\}$, then $\Phi$ has a sequence of critical values $c_k<0$ satisfying $c_k\to 0$ as $k\to \infty$. We improve Clark's theorem by showing that under the assumptions of Clark's theorem $\Phi$ has a sequence of critical points $u_k$ such that $\Phi(u_k)\leq0$ and $u_k\to 0$ as $k\to \infty$. We also generalize Clark's theorem by replacing the $C^1$ smoothness, the boundedness from below, and the (PS) condition with weaker assumptions respectively. The new results produce infinitely many solutions to various nonlinear equations under quite general conditions. (This is joint work with Shaowei Chen and Zhi-Qiang Wang.)