Results about existence/uniqueness of solution to Euler-Lagrange equations? While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading:
What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange equations? Is there any general result? 
I tried to google it, but found nothing.
 A: The  so called   direct method of the  calculus of variations provides one such existence and uniqueness result.
Here is the gist of it.  Suppose that  $X$ is a reflexive  Banach space, e.g.   a Hilbert space or a space of the form $L^p(\Omega)$, $p\in (1,\infty)$, $\Omega$ open subset of some Euclidean space.    We are given a   functional  $J$ on $X$, i.e., a function
$$ J : X\to (-\infty, \infty]$$
and we seek  minimizers of such functionals, i.e.,  points $x_0\in X$ such that
$$J(x_0)=\inf_{x\in X} J(x)$$
The  subset  of $X$ where $J$ is finite is called  the  domain of  $J$.  It is typically described by various equalities and inequalities called constraints.
Existence Theorem. Suppose that $J$ satisfies the following conditions.
\begin{equation}
\inf_{x\in X} J(x)>-\infty.
\tag{A}
\end{equation}
\begin{equation}
 \mbox{The set}\;\;\lbrace
J\leq t\rbrace:=\lbrace x\in X;\;\; J(x)\leq t\rbrace
\;\; \mbox{is convex},\;\;\forall t\in \mathbb{R}.
\tag{B}
\end{equation}
\begin{equation}
 \mbox{The set}\;\;\lbrace
J\leq t\rbrace\;\; \mbox{is closed in the norm topology},\;\;\forall t\in \mathbb{R}.
\tag{C}
\end{equation}
\begin{equation}
\lim_{\|x\|\to\infty} J(x)=\infty.
\tag{D}
\end{equation}
Then   $J$ admits at least one minimizer.
Remark. I should comment on the   four conditions above. Condition (A)  states that $J$ is bounded from below. Condition (B) states that $J$ is a convex function in the usual way.   Condition (C) states that $J$  is lower semicontinuous in the norm    topology. Under the convexity  assumption this is equivalent to $J$ being lower semicontinuous with respect to the  weak topology.  If $J$ happens to be differentiable, then  the differential of $J$ at any minimizer $x_0$ is zero. The ensuing equation $dJ(x_0)=0$   translates into the classical Euler-Lagrange equations.  The minimizer postulated by the above theorem is unique provided that  $J$ is strictly convex. For more about the direct method see  Wikipedia  and the reference therein.
In general, the objects satisfying the Euler-Lagrange equations are critical points of a functional $J: X\to\mathbb{R}$, i.e., points where the differential of $J$ vanishes.      The critical points  that are observable and  detectable  in the real world are stable and these correspond to (local) minimizers of $J$. Sometime, one is interested in   not necessarily stable  objects, i.e., critical points of $J$  that are not necessarily   local minimizers.  Morse theory  is particularly good at detecting    such points.   All applications  of this theory  are based on the following principle.
Suppose that $J: H\to\mathbb{R}$ is a $C^2$ function on a Hilbert space $H$ satisfying some additional compactness assumption (e.g. the Palais-Smale condition).   Suppose that there exist real numbers  $a < b$ such that the sublevel sets  
$$ \lbrace J\leq a\rbrace\;\;\mbox{and}\;\; \lbrace J\leq b\rbrace$$
are not homeomorphic. Then  $J$ admits a  critical point $x_0$ such that
$$ J(x_0)\in [a,b]. $$
For more detail see  the booklet by Paul Rabinowitz, Minimax methods in critical point theory with applications to differential equations.
