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,R§uÀÆNonlinear systems involving curl

6401

9:3010:10

¨«¿§H®ÆAttractors for Kirchhoff wave model with strong damping

6401

10:10-10:30

z10:3011:10

q©²§uÆLiouville-type results for a class of quasilinear elliptic systemsand applications

6401

11:1011:50

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6401

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H_14:3015:10

Ü7§¥IÆêÆXÚÆïÄSome new results on Lane-Emden conjecture, Henon-Lane-Emden conjecture and Schrodinger systems

6401

15:1015:50

Hm²§àHÆSome weighted elliptic equations

6401

15:50-16:10

±t16:1016:40

/¯§u¥ÆOn a class of generalized Monge-Ampere equations

6401

16:4017:10

Û+§u¥ÆLocal uniqueness and applications on nonlinear elliptical equa-tions

6401

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Á ²§¥ìÆWeyl’s law on metric measure spaces

6401

9:009:40

Üá+§¥IÆêÆXÚÆïÄOn the backward uniqueness of parabolic operators

6401

/SX9:4010:20

Congming Li§University of Colorado at Boulder, USAQualitative analysis of solutions to equations with fractionalLaplacian

6401

10:20-10:40

±º10:4011:20

¶ð§¥IÆêÆXÚÆïÄSome results on strongly indefinite variational problems

6401

11:2012:00

H§§¥IÆÉÇÔnêÆïĤGround states of two-component attractive Bose-Einstein con-densates

6401

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Some results on strongly indefinite variational problems

¶ð§¥IÆêÆXÚÆïÄ

Consider the following general nonlinear system Au = N(u) where H is a Hilbert space,

A is a self-adjoint operator, and N is a (nonlinear) gradient operator. Typical example are

Dirac equations and reaction-diffusion systems where σ(A) (the spectrum) is unbounded

from below and above, and particularly, σe(A)⋂

R± 6= ∅. The talk focus on applications to

Dirac equation and the reaction-diffusion systems, etc..

Ground states of two-component attractive Bose-Einstein

condensates

H§§¥IÆÉÇÔnêÆïĤ

This talk is focussed on the ground states of two-component Bose-Einstein condensates

(BEC) with trapping potentials in R2, where the intraspecies interaction and the interspecies

interaction are both attractive. The existence and non-existence of ground states are classi-

fied completely by investigating equivalently the associated L2-critical constraint variational

problem. Some results on the uniqueness and symmetry-breaking of ground states are also

discussed.

Some weighted elliptic equations

Hm²§àHÆ

Structure of solutions of some weighted second and fourth order elliptic equations are

studied via Morse index and new embeddings of weighted Sobolev spaces. The structure of

entire solutions are classified via Morse index and some Liouville type results are established

for the supercritical case. The existence and regularity of solutions of the Dirichlet problems

in bounded smooth domains and the related Liouville type result are also obtained for the

subcritical case.

14 u¥Æ

Qualitative analysis of solutions to equations with fractional

Laplacian

Congming Li§University of Colorado at Boulder, USA

We present some recent work on the study of nonnegative solutions to nonlinear equa-

tions with fractional Laplacian. We focus on Liouville type theorem, the symmetry and

monotonicity of solutions. The main tools are some forms of maximum principles which are

‘much better’ than in the Laplace case. We also present some Bocher type theorems. These

Bocher type theorems are used to derive maximum principles on a punctured ball.

Local uniqueness and applications on nonlinear elliptical equations

Û+§u¥Æ

In this talk, we give the uniqueness of positive solutions with concentration phenomenon

to nonlinear Schrodinger equation and the Brezis-Nirenberg problem. Then we get the

number of positive solutions under some conditions. These are mainly depend on the local

Pohozaev identity and blow-up analysis. These are the work jointed with Daomin Cao,

Shuanglong Li and Shuangjie Peng.

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þO5£ã)½)Y²8à5. ·0AïÄd¯K,¿ÛA

a,¯K.

Nonlinear systems involving curl

,R§uÀÆ

In this talk we shall present recent results on solvability and regularity of solutions of

nonlinear differential systems involving curl. Solvability of such systems depends on the

nature of nonlinearity of the equations and the type of the boundary conditions, and very

often depends also on the domain topology. Some open problems will be presented.

u¥Æ 15

On a class of generalized Monge-Ampere equations

/¯§u¥Æ

We give comparison principle for weak solutions to a class of generalized Monge-Ampere

equations.

On the backward uniqueness of parabolic operators

Üá+§¥IÆêÆXÚÆïÄ

We consider the backward uniqueness of general parabolic operators of divergent type

in the whole space. Under the assumptions of boundedness of scaling invarant gradient

of the coefficients, we proved the backard uniqueness for solutions at most exponentially

quadratic growth. This is a jointed work with Wu Jie.

Some new results on Lane-Emden conjecture, Henon-Lane-Emden

conjecture and Schrodinger systems

Ü7§¥IÆêÆXÚÆïÄ

By Sobolev embeddings on SN−1 and the scaling invariant of the solutions, we show

Lane-Emden conjecture holds in a new region; and we prove Henon-Lane-Emden conjecture

is true for space dimension N = 3.

We also show new results on some Schrodinger systems, such as symmetry and asymp-

totic behavior of ground state solutions; bifurcation and multiple existence of positive solu-

tions; uniqueness and structure of positive solutions; symmetry breaking via Morse index.

Attractors for Kirchhoff wave model with strong damping

¨«¿§H®Æ

The long-time behavior of Kirchhoff equations is investigated. We first present the

well-posedness and existence of global attractor for Kirchhoff type wave equations in regular

space, where we assume that the stiffness coefficient is non-degenerate. Our second result

deals with a global attractor which attracts every H10 (Ω)×L2(Ω)-bounded set with respect

to the H10 (Ω)×H1

0 (Ω) norm. Finally, we give the existence of global attractor for Kirchhoff

wave equations in natural energy space in the case when the stiffness coefficient is degenerate.

16 u¥Æ

(Jointly with Honglu Ma).

Weyl’s law on metric measure spaces

Á ²§¥ìÆ

The classical Weyl’s law gives the asymptotic formula of Dirichlet eigenvalues. It has a

wide range of interests about the extensions of this classical result. In this talk, I will report

such an extension to non-smooth settings. This is a joint work with Hui-Chun Zhang.

Liouville-type results for a class of quasilinear elliptic systems and

applications

q©²§uÆ

In this talk, we study the quasilinear elliptic systems with the m-Laplacian operator

(m > 1). Under appropriate conditions on the functions, we prove some Liouville-type

nonexistence theorems. The Liouville-type results can be applied to non-cooperative quasi-

linear Schrodinger-type systems, and we obtain a priori estimates, singularity and decay

estimates for the nonnegative solutions of the non-cooperative quasilinear Schrodinger-type

systems. (Jointly with N. Dancer and H. Yang).

u¥Æ 17

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Li

University of Colorado at Boulder,

USA

Ç congmingli@gmail.com

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