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.     

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.     

非线性湿气迁移方程的超收敛分析

本文用混合有限元方法研究一般的非线性湿气迁移方程.利用双线性元Q₁₁和零阶Raviart-Thomas元(Q₁₀×Q₀₁)证明方程的超收敛性.利用这两个单元插值算子的性质和平均值技巧,得到了方程半离散格式的O(h²)阶超收敛结果.对于方程线性化的Crank-Nicolson(C-N)全离散格式,得到了具有O(h²+τ²)阶的超收敛结果,这里h是空间剖分参数,τ是时间步长.该方法说明如果线性化问题有超收敛性,那么对应的非线性问题有同样的超收敛性.最后,给出数值算例,证实了理论分析的正确性和方法的有效性.     

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.     

一类有Hardy-Sobolev临界指数的椭圆方程的多解性

证明了具有Hardy-Sobolev临界指数的半线性椭圆方程(1)的解的情况,存在λ*>0,当λ∈(0,λ*)时,运用对偶喷泉定理得方程有无穷多解,且该解序列具有负的能量值;当λ→0 时,解的模趋于零;当λ≤0时,方程没有负能量的解.     

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.