Speaker: Prof.Zhaoli Liu（Capital Normal University）

Title: Improvements and generalizations of Clark's theorem and applications

Time: Sat, Nov.23 2019,AM:10:00-11:00

Location: Room 3A02, Building No.37, Wushan Campus

Abstract：

     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.)