I am searching for (semi)-analytical methods to approach the following non-linear boundary value problem
\begin{align*} y''(x) - \lambda e^{y(x)} - \alpha&=0, \qquad \lambda,\alpha>0 \\ y(0)=y(1)=0 \end{align*} I am particularly interested in the case where $\lambda$ is large. I believe an explicit analytic solution does not exist. I can find an implicit solution by reducing the order via $y'(x) = \omega(y(x)) = \pm\sqrt{\lambda e^{y}+\alpha y+c_1}$ where $c_1$ is an integration constant, but it would be helpful to have an explicit approximation.
I have tried approaching the problem with the Adomian decomposition, but since in the end I am interested on large values of $\lambda$, the approximation (a polynomial proportional to $\lambda$) does not seem to converge well to the solution.
I am not sure if this helps, but the related problem with $\alpha=0$ (a.k.a Liouville problem) can be solved exactly by \begin{align*} y_{\alpha=0}(x) = 2\log\left[\frac{\cos{\frac{\theta}{4}}}{\cos\left(\frac{\theta}{2}\left(\frac{1}{2}-x\right)\right)}\right] \end{align*} where the parameter $\lambda$ is related to $\theta$ via the equation $\theta = \sqrt{2\lambda} \cos{\frac{\theta}{4}}$. Here $\theta=2\pi$ corresponds to the asymptotic limit of interest $\lambda \to \infty$. Related to my remark above, $y_{\alpha=0}(x)$ reaches its maximum at $x=1/2$ where $y_{\alpha=0}(1/2)=2\log{\cos{\frac{\theta}{4}}}$, diverging logarithmically as $\theta\to2\pi$, so an approximation of $y_{\alpha=0}$ given by a power series proportional to $\lambda$ diverges faster.
Any ideas?