Necessary and Sufficient Conditions for the Solvability of a Nonlinear Two-Point Boundary Value Problem
J. Mawhin, J. R. Ward, M. Willem · Proceedings of the American Mathematical Society · 1985
The dual least action principle is used to prove a necessary and sufficient condition for the solvability of a Dirichlet problem of the form $u'' + u + f\left ( {x,u} \right ) = 0$. $u(0) = u(\pi ) = 0$ when $f\left ( {x, \cdot } \right )$ is nondecreasing and $\int _0^u {f\left ( {x,v} \right )dv}$ satisfies a suitable growth condition.