# Steady state equilibrium in Solow model with a convex production function

Suppose an economy is producing $$e^k$$ amount of output per capita if it uses $$k$$ amount of capital per capita.

As the production function is strictly convex I am thinking the only steady state is at $$k=0$$; hence the answer should be that 'this model may not have a steady state equilibrium'.

Can someone please let me know if my thinking is correct?

You only provide partial information. E.g., this production function is unusual; is anything else unusual? Is depreciation still linear in $$k$$? Is the rate of population growth constant? etc.
If nothing else is unusual: you can repeat the usual steps for finding a steady state, find if a $$k$$ exists for which per capita savings/investment equal per capita depreciation: $$s \cdot e^k = \delta \cdot k \tag{1}$$ Note that $$k = 0$$ does not work as you have $$s \cdot e^0 = s \neq 0 = \delta \cdot 0.$$
Note that convexity only guarantees that any interior equilibrium you may find would be unstable; increasing (decreasing) $$k$$ will increase (decrease) the LHS of (1) more than the RHS of (1), thus a positive (negative) shock will be further reinforced by increased (decreased) $$k$$ levels.