Special Session 111: Partial Differential Equations and Material Sciences

An elliptic problem with periodic boundary condition involving critical growth

Yuxia GUO Tsinghua University Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will talk about an elliptic problem involving critical growth in a strip, satisfying the periodic boundary condition.As a consequence, we prove that the prescribed scalar curvature problem in $\mathbb R^N$ has solutions which are periodic in some variables, if the scalar curvature $K(y)$ is periodic.

Bruun-Minkowski Inequalities in Some Variational Problems

Meiyue Jiang School of Mathematical Sciences, Peking University Peoples Rep of China Co-Author(s):    Meiyue Jiang

Abstract: In this talk, motivated by the Brunn-Minkowski inequalities in convex geometry, we will discuss the Brunn-Minkowski type inequalities in variational problems related to some nonlinear elliptic equations.

Properties of ground states for two-component attractive Bose-Einstein condensates

Xiaoyu Zeng Wuhan University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we study a system of two coupled time-independent Gross-Pitaevskii equations, which is used to model two-component Bose-Einstein condensates with attractive interactions. For a certain type of trapping potentials, especially for the degenerate ring-shaped potentials, we investigate the existence, concentration and local uniqueness of ground states.

Optimal higher derivative estimates for solutions of the Lam\`e system with closely spaced hard inclusions

Peihao Zhang Beijing Normal University Peoples Rep of China Co-Author(s):    Hongjie Dong; Haigang Li; Huaijun Teng

Abstract: We study the higher derivative estimates for the Lam\`e system with hard inclusions embedded in a bounded domain in $\mathbb{R}^{d}$. The stress in the narrow regions between two closely spaced hard inclusions significantly increases as $\varepsilon$, the distance between inclusions, approaches to 0. The stress is represented by the gradient of the solution. The novelty of this paper is that to fully characterize this singularity, we derive higher derivative estimates for solutions to the Lam\`e system with partially infinite coefficients. These upper bounds are shown to be sharp in dimensions two and three when the domain exhibits certain symmetry. To the best of our knowledge, this work is the first to quantify precisely this singular behavior of the higher derivatives for the Lam\`e system with hard inclusions.

Some results on Kirchhoff type elliptic equation on $\mathbb R^N$

Huan-Song Zhou Wuhan University of Technology Peoples Rep of China Co-Author(s):    H L Guo, X Y Zeng, Y M Zhang, L F Weng and X Zhang

Abstract: We are concerned with certain nonlinear Schrodinger equations involving Kirchhoff type nonlocal term. The existence and concentration behavior of $L^2-$normalized solutions for the equations are discussed. Moreover, due to the presence of Kirchhoff type nonlocal term in the equations, it is well known that some strong growth conditions on the nonlinear terms of the equations are required when we seek a mountain pass solution (i.e., a solution obtained by using mountain pass theorem), in this talk we will show that there is a way to get a mountain pass solution for a Kirchhoff type elliptic equation under the usual growth condition on the nonlinear term. The work was supported by NSFC.