带深井位势双调和方程的解

本文研究下述双调和方程极小能量解的存在性:?~2u+[λV (x)-δ]u=|u|_(p-2)u, x∈R~N,(0.1)其中N≥5,λ 0. p是次临界或临界的Sobolev指标,即2 p≤2**,这里2**=2N/N-4为临界的Sobolev指标, V (x)是非负连续的深井位势,其零集V~(-1)(0):={x∈R~N:V (x)=0}的内部int V~(-1)(0)是R~N中非空的有界光滑区域.令μ0为定义在int V~(-1)(0)中齐次边界条件下?~2的第一特征值.对任意的0 δμ0,本文证明:当λ 0充分大时,(0.1)存在一个在V~(-1)(0)附近的极小能量解.     

一类涉及分数阶Hardy-Schrödinger算子和Hardy-Sobolev临界指数的方程组

本文研究一类涉及分数阶Hardy-Schr?dinger算子和Hardy-Sobolev临界指数的方程组.在适当的条件下,本文获得该方程组基态解的存在性和相关性结果.为克服紧性缺失,本文考虑一个定义在有界区域上的次临界辅助问题,并证得该辅助问题对应的紧嵌入性结论.     

一类含梯度项的二阶半线性椭圆型方程的爆炸解

本文运用非线性变换、摄动方法结合上、下解方法得到了一类含梯度项的二阶半线性椭圆型方程爆炸解的存在性．     

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS

The author first analyzes the existence of ground state solutions and cylindrically symmetric solutions and then the asymptotic behavior of the ground state solution of the equation -△u=φ(r)up-1,u>0 in RN, u ∈ D1,2(RN),where N≥ 3,x = (x',z)∈ RK×RN-K,2≤K≤N,r =|x'|.It is proved that for 2(N -s)/(N-2) < p < 2* = 2N/(N -2),0 < s < 2, the above equation has a ground state solution and a cylindrically symmetric solution. For p=2*, the above equation does not have a ground state solution but a cylindrically symmetric-solution, and when p close to 2*, the ground state solutions are not cylindrically symmetric. On the other hand, it is proved that as p close to 2*, the ground state solution up has a unique maximum point xp = (x'p,zp) and as p→2*, |x'p|→r0 which attains the maximum of φ on RN.The asymptotic behavior of ground state solution up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2*.     

一类带Hardy项的椭圆方程的无穷多解

假设 $\Omega=B_R:=\{x\in \mathbb{R}^N:|x|0$, $ N \geq 7$, $ 2^*=\frac{2N}{N-2}$, 我们得到了如下半线性问题无穷多解的存在性: $\left\{ \begin{array}{ll} -\Delta u=\frac{\mu}{|x|^2}u+|u|^{2^*-2}u+\la u, &; x\in\Omega, \\ u=0, &; x\in \partial\Omega. \end{array} \right.$ 其中$\lambda \in \mathbb{R}, \mu \in \mathbb{R}$. 这些解由不同的节点来区分.