According to Prof. Israel's database, fuctions to be evaluated with allsolve should be twice continuously differentiable in their range. That condition probably doesn't hold for the equation describing P. Unlike the tan(x)=x example the actual trancendental is unknown to me (it's somewhere in the 'black box' of fsolve(eqs)) and even if I knew it's form I doubt I would be able to transform it into functions that *are* twice continuously differentiable. 

I more or less decided to asquiesce in using trial and error for finding the smallest value. When I have more time I will certainly try out myself the things Axel tried.

Thanks again, everybody.

Regards, Erik

Thanks for all the suggestions, Axel!

I am now looping your iterative scheme (while (true) do), and when I give it enough time it can find the minimum value
Iterating is probably the only way to find all solutions in a range and pick the smallest.
P is given by a transendental function, for example tanx=x:

Apparently fsolve can find the solution for only one of these curves in range, otherwise op(%) should have had more than one entries.

I'm happy with not having to rerun the file time after time:)

Regards, Erik

Shouldn't solving for 10-P on a range 0..10 give the same solution as solving -P on a range -10..0?

Of course it's always possible to find the smallest value by running the file several times and putting the upper bound of the range closer to 0, until maple can no longer find a solution. It's just rather time consuming (on my pc). I included the file this time.

View 8135_buckling4.mw on MapleNet or Download 8135_buckling4.mw
View file details

I've tried solving for -P on a range of -10..0 (I guess that's equal to your tip)
I'm afraid it doesn't work, fsolve then returns the most negative solution. What I'm looking for is the solution closest to 0.

But thanks for thinking along

Thanks Georgios,

I was really amazed when I saw fsolve had provided an actual answer, and when I ran your file it gave 'error: too many recursions'.

It turns out Maple 9.5 at home wasn't able to evaluate the problem, but Maple 11 at university was.  Apparently Maple 11 has some implicit setting for fsolve to find the smallest solution >0 when there are infinitely many solutions. 

Regards, Erik