I've tried solving the following (diffusion) equation numerically ($\alpha$ is const)
$$\frac{1}{x}\frac{d}{dx}\left(\frac{x}{T^{1/2}}\frac{dT}{dx}\right)+\alpha T^{3/2}=0$$
since it has no analytic solution. The BC's are $T(1)=0$ and $T'(0)=0$. The numerical solution points out to $T(x)=0$ as the solution.
I also tried an equivalent solution by setting $V=T^{1/2}$:
$$\frac{1}{x}\frac{d}{dx}\left(x\frac{dV}{dx}\right)+\frac{\alpha}{2}V^3=0,$$
but I'm also getting $V(x)=0$.
The problem is that I'm quite convinced that this has other solution than the trivial one (basically by physical arguments).
Is there any way of prooving that another solution exists besides the trivial one?
NOTE: I've seen in a book the author assuming a certain profie for $T(x)$ with some free parameters, and then taking the first moments of the equation to get those free parameters. I am not however very confident with that approach.
Consider the problem given by $$ \frac{\partial T}{\partial t} = \frac{1}{x} \frac{\partial}{\partial x} \left( x k(T) \frac{\partial T}{\partial x} \right) + f(T) $$ $$ T = T_0(x), \ \ \ t=0 $$ $$ \frac{dT}{dx} = 0, \ \ \ x=0 $$ $$ T = 0, \ \ \ x=1 $$
This problem describes the transient heat transfer along the radial direction of an infinitely long cylinder of radius $1$. The surface of that cylinder is kept at temperature $0$. Imagine initially that there is no heat generation (i.e., $f(T) = 0$). In that case, since the surface temperature cannot change, the cylinder will lose heat to the region surrounding the cylinder until the temperature of the cylinder is also $0$ (which should normally take an infinite amount of time). Therefore, the steady-state solution (corresponding to the case with $\partial T/\partial t = 0$) will be $T=0$.
However, if there is heat generation ($f(T)\neq 0$) and the initial temperature profile $T_0(x)$ is nonzero, the cylinder could keep itself heated if "enough" heat is generated. Therefore, even losing heat to the ambient the cylinder would still have a nonzero temperature distribution due to the heat source, which itself depends on the temperature. Naturally, if $f(0)=0$ and the initial temperature profile is $T_0(x)=0$, the cylinder will not be able to heat itself and the solution will be $T=0$ for all $t>0$ and $0<x<1$.
In your case, $k(T) = T^{-1/2}$ and $f(T)=\alpha T^{3/2}$, which corresponds to the case with $f(0)=0$. Therefore, in the alternative problem of transient heat conduction, the steady-state solution can depend on the initial profile. A numerical approach to this issue could involve the solution of the transient PDE instead of the steady-state ODE and verify the dependence of the steady-state solution on the initial profile. That kind of PDE is usually not more difficult to solve than some ODE's. Take a look at the FTCS or Crank-Nicholson schemes.
I previously commented about using the Picard-Lindelöf theorem to study the uniqueness of the solution of your problem. Later I noticed that your problem is a boundary value problem, and not a initial value problem (which is the kind of problem to which the theorem applies). Therefore, a rigorous mathematical study of the uniqueness of the solution will be more complicated.