The Rabinowitz minimal periodic solution conjecture on partially convex reversible Hamiltonian systems and brake subharmonics

Under weaker regularity and compactness assumptions, we find the mountain-pass essential point, which is a novel extension of the classical Ambrosetti-Rabinowitz mountain pass theorem. We study the reversible superquadratic autonomous Hamiltonian systems whose Hamiltonian $H(p,q)$ is strictly convex in the position $q\in\mathbf{R}^n$ and prove that for every $T>0$, the system has a $T$-periodic brake solution $\bar x$ with minimal period $T$, provided the Hessian $H_{pp}(\bar x(t))\in\mathbf{R}^{n\times n}$ is semi-positive definite for $t\in\mathbf{R}$ or $n=1$. For brake subharmonics of general reversible nonautonomous Hamiltonian systems, we also get some new results.