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.
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.
