I am studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I am looking for a reference of three following theorems:
Let $f(x)$ be a convex funtional defined in a open convex set of a normed space $E$. Let $f(x)$ be a real Gateaux differentiable functional with $f'(x,\cdot )$ continuous for fixed $x\in E$. Then $f(x)$ is weakly lower semi-continuous.
If a strictly convex functional $f(x)$ defined in a linear space $E$ has a minimum at a point $x_0$, then $x_0$ is an absolute minimum point, and there are no other minimum points.
Let $f(x)$ be a real Gateaux differentiable funtional defined in a real reflexive Banach space $E$, which is weakly lower semi-continuous and satisfies the condition $f'(x,x)>0$ for any vector $x$ in $E$, $|x|=R>0$ and $f'(x)=gradf(x)$. Then there is exists an interior point $x_0$ of the ball $|x|\leq R$ at which $f(x)$ has a local minimum so that $f'(x)=0$.
Taubes paper cites the book 'Variational method and method of monotone operators in the theory of nonlinear equations' by M. M. Vainberg. However, the library in my university does not have this book. Is there another reference where I can find these results?
Thanks.
