Use Lagrange multipliers as applied in optimization of nonlinear function.

I don't know what you mean by L=1, but the key issue here is which, if any, of the two constraints are binding. If you have strictly positive prices for the two variables, your production function constraint will be binding.  To find out which case you are dealing with, set up the Lagrangean with the production function constraint equation and solve the first order conditions for the minimization in the usual way. Now see if your proposed solution satisfies the second constraint. i.e. whether the second constraint is slack. If your proposed solution also satisfies the second constraint then you have found your optimum. If your proposed solution does not satisfy the second constraint then the second constraint is also binding at the optimum. In this case the solution to the problem is given by the  solution to the pair of equations for your constraints.