临界双调和方程非平凡解的存在性

在有界光滑区域Ω∈R~N(N4)上,研究双调和方程△~2u-λu=|u|~(2_*-2)u,x∈Ω,u=(δu)/(δn)=0,x∈δΩ,其中2_*=2N/(N-4)是临界指数.对于任意的λ0,利用变分方法可以得到上面方程非平凡解的存在性.     

一类双调和方程的多解性

本文研究了非线性项在无穷远处次线性增长的一类双调和方程解的存在性和多解性.应用抽象临界点定理,证明了此类双调和方程至少有三个弱解存在.     

STABILITY OF GASEOUS STARS IN THE NON-ISENTROPIC CASE

The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this article is concerned with the nonlinear stability of gaseous stars in the non-isentropic case, when 34 γ2, S(x,t) is a smooth bounded function. First, we verify that the steady states are minimizers of the energy via concentration-compactness method; then using the variational approach we obtain the stability results of the non-isentropic flow.     

MULTIPLE STATIONARY SOLUTIONS OF EULER-POISSON EQUATIONS FOR NON-ISENTROPIC GASEOUS STARS

The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for some velocity fields and entropy functions that solve the conservation of mass and energy. Under different restriction to the strength of velocity field, we get the existence and multiplicity of the stationary solutions of Euler-Poisson system.     

Remarks on the nonlinear instability of incompressible Euler equations

In this paper, we consider the nonlinear instability of incompressible Euler equations. If a steady density is non-monotonic, then the smooth steady state is a nonlinear instability. First, we use variational method to find a dominant eigenvalue which is important in the construction of approximate solutions, then by energy technique and analytic method, we obtain the dynamical instability result.