2021.11.13（星期六）上午900-1000

点：腾讯会议：273 279 347

Estimate, Existence and nonexistence of positive solutions of Hardy-H\'{e}non equations

要：We consider the boundary Hardy-H\'{e}non equation $-\Delta u=(1-|x|)^\alpha u^p,\ \ x\in B_1(0)$, where $B_1(0)\subset\mathbb{R}^N$ $(N\geq 3)$ is a ball of radial $1$ centered at $0$, $p>0$ and $\alpha\in \mathbb{R}$. We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case $0<p<\frac{N+2}{N-2}$, we establish the estimate of positive solutions. When $\alpha\leq -2$ and $p>1$, we give some conclusions with respect to nonexistence. When $\alpha>-2$ and $1<p<\frac{N+2}{N-2}$, we obtain the existence of positive solution for the corresponding Dirichlet problem. When $0<p\leq 1$ and $\alpha\leq -2$, we show the nonexistence of positive solutions. When $0<p<1$, $\alpha>-2$, we give some results with respect to existence and uniqueness of positive solutions.

Hopf bifurcation from spike solutions for some Turing reaction diffusion systems

要：Turing systems are classic reaction-diffusions systems  modeling patterned solutions resulted from Turing instability. The stability problem of spike solution leads to Hopf bifurcation. We prove the existence of Hopf bifurcation from the spike solutions using the classic Crandall-Rabinowitz Hopf bifurcation Theorem. The stability of the bifurcating periodic solution is also studied using a combined theoretic and numerical methods.(Joint work with D. Gomez and J.C.Wei.)

：带有混合非线性项的分数阶薛定谔方程正规化解的存在性和稳定性

要：我们考虑一类带有混合非线性项的分数阶薛定谔方程，研究其在给定质量下解的存在性和稳定性。具体说来，对于L^2次临界情形，我们利用Lions集中紧性原理证明解的存在性以及轨道稳定性。对于L^2超临界情形，利用山路定理证明解的存在性，另外，我们构造了几种等价的变分刻画来研究解的强不稳定性。而对于同时具有L^2超临界和L^2次临界非线性项，研究其解的存在性就复杂得多，我们通过对其Pohozaev流形进行分解来构造合适的环绕结构，从而通过一般的临界点定理证明了解的存在性，此外，我们也证明了某些情形下解的强不稳定性。