In this paper, we study the existence of multiple solutions for the boundary value problem

$$\begin{aligned} -\Delta _{\gamma } u =& f(x,u) + g(\theta ,x,u) \ \text{ in }\ \ \Omega , \\ u =&\ 0 \hspace{2.6cm}\text{ on }\ \partial \Omega , \end{aligned}$$

where \(\Omega \) is a bounded domain with smooth boundary in \(\mathbb{R}^{N} \ (N \ge 2)\), where \(f(x,\cdot )\) is odd, \(g(\theta ,x, \xi )\) is a non-symmetric, perturbative term and \(\Delta _{\gamma } \) is the strongly degenerate elliptic operator of the type

$$ \Delta _{\gamma }: =\sum _{j=1}^{N}\partial _{x_{j}} \left ( \gamma _{j}^{2} \partial _{x_{j}} \right ), \quad \partial _{x_{j}}: = \frac{\partial }{\partial x_{j}},\quad \gamma : = (\gamma _{1}, \gamma _{2}, \ldots, \gamma _{N}). $$

By using a perturbation method introduced by Bolle (J. Differ. Equ. 152:274-288, 1999), we prove the existence of multiple solutions in spite of the lack of symmetry of the problem.

中文翻译：

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缺乏对称性的半线性强退化椭圆微分方程的无穷多解

本文研究边值问题多解的存在性

$$\begin{aligned} -\Delta _{\gamma } u =& f(x,u) + g(\theta ,x,u) \ \text{ in }\ \ \ Omega , \\ u =& \ 0 \hspace{2.6cm}\text{ on }\ \partial \Omega , \end{aligned}$$

其中\(\Omega \)是在\(\mathbb{R}^{N} \ (N \ge 2)\) 中具有平滑边界的有界域，其中\(f(x,\cdot )\)是奇数, \(g(\theta ,x, \xi )\)是一个非对称的微扰项，而\(\Delta _{\gamma } \)是该类型的强退化椭圆算子

$$ \Delta _{\gamma }: =\sum _{j=1}^{N}\partial _{x_{j}} \left ( \gamma _{j}^{2} \partial _{x_ {j}} \right ), \quad \partial _{x_{j}}: = \frac{\partial }{\partial x_{j}},\quad \gamma : = (\gamma _{1}, \gamma _{2}, \ldots, \gamma _{N})。$$

通过使用 Bolle (J. Differ. Equ. 152:274-288, 1999) 引入的扰动方法，尽管问题缺乏对称性，但我们证明了多个解的存在。