关于验证指定函数是否为带边界条件的椭圆微分方程解的技术问询

Hey everyone, I'm currently trying to verify that the following functions satisfy a given elliptic partial differential equation (PDE) paired with boundary conditions. This problem comes from Dupaigne's Stable solutions of elliptic partial differential equations, page 34, and I wanted to share my attempt so far while asking for any guidance or checks on my work.

First, here are the candidate solutions we're examining:
$$
u_\lambda(r)=\ln \frac{8 b_{-}}{\left(1+\lambda b_{-} r^2\right)2}, \quad U_\lambda(r)=\ln \frac{8 b_{+}}{\left(1+\lambda b_{+} r^2\right)2}
$$
where the coefficients are defined as:
$$
b_{ \pm}=\frac{4-\lambda \pm \sqrt{16-8 \lambda}}{\lambda^2}, \quad r \in[0,1]
$$

The PDE we need to verify (for the radial variable $r$, where $B$ denotes the unit ball in $\mathbb{R}^2$) is:
$$
-\frac{d^2u}{dr2} = \lambda e^u \quad \text{in } B
$$
along with the boundary condition:
$$
u=0 \quad \text{on } \partial B
$$

Since the verification procedure is nearly identical for both $u_\lambda(r)$ and $U_\lambda(r)$, I was planning to work through one of them in detail to demonstrate my approach. I'd love to get input on whether I'm framing this correctly, or if there are key steps I might be overlooking as I proceed!

备注：内容来源于stack exchange，提问作者Richard