我们证明了存在性，非退化，并在无穷远的非平凡的解决方案埃姆登的方程$-Δu = | u |^3$上的一个无界的$L$形域，Dirichlet边界条件的指数衰减。除了这个结果的直接价值，我们也把这个解决方案作为一个积木的解决方案扩大有界域的角落，建立在未来的工作。我们的证明大量使用计算机辅助：从一个数值近似解开始，我们使用一个不动点参数来证明附近的精确解的存在性。在证明过程中建立的特征值界也意味着解的非退化性。
英文摘要：
We prove existence, non-degeneracy, and exponential decay at infinity of a non-trivial solution to Emden's equation $-Δu = | u |^3$ on an unbounded $L$-shaped domain, subject to Dirichlet boundary conditions. Besides the direct value of this result, we also regard this solution as a building block for solutions on expanding bounded domains with corners, to be established in future work. Our proof makes heavy use of computer assistance: Starting from a numerical approximate solution, we use a fixed-point argument to prove existence of a near-by exact solution. The eigenvalue bounds established in the course of this proof also imply non-degeneracy of the solution.
Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term
研究了奇异椭圆型方程$-Δu= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$（$N\geq 3$），其中$p$为正权，$0< a <1$为正权.在假设$f$是次线性增长的非减函数，而$g$是减函数且在原点附近无界的条件下，我们建立了在无穷远处消失的基态解的存在性.我们的论证基本上依赖于最大值原理。
Method of scaling spheres: Liouville theorems in inner or outer (unbounded or bounded) generalized radially convex domains, blowing-up analysis on domains with not necessarily $C^1$ boundary and other applications
