论文标题
椭圆形的非线性和电势在起源和域边界处具有单数
Elliptic problems with mixed nonlinearities and potentials singular at the origin and at the boundary of the domain
论文作者
论文摘要
我们对以下dirichlet问题$$ \ left \ weet {ar I {array} {ll}-Δu +λu -μ\ frac { f（x，u）＆\ quad \ mbox {in}ω\\ u = 0＆\ quad \ mbox {on} \ partialω，\ end end {array} \ right。有界域上的$$ $ω\ subset \ mathbb {r}^n $，$ 0 \ inω$。我们假设非线性零件在某些封闭的子集$ k \ subsetω$上是超级线性，并且在$ω\ setMinus k $上渐近线性。我们找到了一种具有特定最小值水平的能量的解决方案，并且只要$ f $在$ u $中是奇怪的。此外，我们还研究了针对相关标准化问题的多种解决方案。
We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -Δu + λu - μ\frac{u}{|x|^2} - ν\frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus Ω)^2} = f(x,u) & \quad \mbox{in } Ω\\ u = 0 & \quad \mbox{on } \partial Ω, \end{array} \right. $$ on a bounded domain $Ω\subset \mathbb{R}^N$ with $0 \in Ω$. We assume that the nonlinear part is superlinear on some closed subset $K \subset Ω$ and asymptotically linear on $Ω\setminus K$. We find a solution with the energy bounded by a certain min-max level, and infinitely many solutions provided that $f$ is odd in $u$. Moreover we study also the multiplicity of solutions to the associated normalized problem.
