Non-existence criteria for Laurent polynomial first integrals

In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial first integrals for a general nonlinear systems of ordinary differential equations $\dot x = f(x)$, $x \in \mathbb{R}^n$ with $f(0) = 0$. We show that if the eigenvalues of the Jacobi matrix of the vector field $f(x)$ are $\mathbb{Z}$-independent, then the system has no nontrivial Laurent polynomial integrals

Existence of nontrivial solutions to a fourth-order Kirchhoff type elliptic equation with critical exponent

In this paper, a critical fourth-order Kirchhoff type elliptic equation with a subcritical perturbation is studied. The main feature of this problem is that it involves both a nonlocal coefficient and a critical term, which bring essential difficulty for the proof of the existence of weak solutions. When the dimension of the space is smaller than or equals to $7$, the existence of weak solution is obtained by combining the Mountain Pass Lemma with some delicate estimate on the Talenti's functions. When the dimension of the space is larger than or equals to $8$, the above argument no longer works. By introducing an appropriate truncation on the nonlocal coefficient, it is shown that the problem admits a nontrivial solution under appropriate conditions on the parameter