Let $X$ be a Banach space and consider the following minimisation problem: $$ \inf_{u\in X} \int_\Omega f(x, u(x), \nabla u(x))\, dx \ \ \text{ subject to } \ \ \int_\Omega g(x, u(x))\, dx = 0, $$ where $f, g$ are assumed to be continuously differentiable and have appropriate growth bounds so that the problem has a minimiser and one can write down the associated Euler-Lagrange equation. For the unconstrained problem, I know that a solution to the Euler-Lagrange equation is a minimiser if one assumes that $(v, A)\mapsto f(x, v, A)$ is jointly convex (together with a stronger growth bound on $f$).
For the constrained problem, are there any conditions that one can impose on $f$ and $g$ that give the same result? It would be great if you can provide a reference as well.