Euler-Lagrange equation and unknown coefficients

Question

I want to show that the nonlinear functional

$$ J(u) = \int_0^1 (u'(x))^2 + b(x)u^2(x) + f(x) u(x) \,\textrm{d}x $$

attains its minimum in exactly one point of the Sobolev space $W_0^{1,2}(0,1)$.

What I would like to do is to write the Euler-Lagrange equations for that functional. Will it be sufficient to show the uniqueness?

$$\frac d {dx} L_p(x, u, p) = L_u(x, u, p),$$

where $p = u'$. The equation in my case is (after some simple cancellations) $2u''(x) - 2u(x)b(x) - f(x) = 0$.

Answer 1

The uniqueness follows from:

• every minimizer of $J$ satisfies the Euler-Lagrange equations
• the Euler-Lagrange equations have a unique solution

If you can show both facts, you are done.