We are interested in the following Dirichlet problem:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\textrm{dist}(x,\mathbb {R}^N \setminus \Omega )^2} = f(x,u) &{} \quad \text{ in } \Omega \\ u = 0 &{} \quad \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$

on a bounded domain \(\Omega \subset \mathbb {R}^N\) with \(0 \in \Omega \). We assume that the nonlinear part is superlinear on some closed subset \(K \subset \Omega \) and asymptotically linear on \(\Omega \setminus K\). 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.

中文翻译：

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在原点和域边界处具有混合非线性和势奇异的椭圆问题

我们对以下狄利克雷问题感兴趣：

$$\begin{对齐} \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u} {\textrm{dist}(x,\mathbb {R}^N \setminus \Omega )^2} = f(x,u) &{} \quad \text{ in } \Omega \\ u = 0 &{ } \quad \text{ 上 } \partial \Omega 、 \end{array} \right。\end{对齐}$$

在有界域\(\Omega \subset \mathbb {R}^N\)和\(0 \in \Omega \)上。我们假设非线性部分在某些闭合子集\(K \subset \Omega \)上是超线性的，并且在\(\Omega \setminus K\)上是渐近线性的。我们找到一个能量以某个最小–最大水平为界的解，并且只要f在u中是奇数，就有无限多个解。此外，我们还研究了相关标准化问题的多种解决方案。