First we examine a resonant variational inequality driven by the $p$-Laplacian and with a nonsmooth potential. We prove the existence of a nontrivial solution. Then we use this existence theorem to obtain nontrivial positive solutions for a class of resonant elliptic equations involving the $p$-Laplacian and a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory for functionals of the form $\varphi ={\varphi }_1+{\varphi }_2$ with ${\varphi }_1$ locally Lipschitz and ${\varphi }_2$ proper, convex, lower semicontinuous.

A $1$-harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in ${ℝ}^N$, is formulated by the use of subdifferentials of a singular energy—the total variation. An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result, a local-in-time solution of $1$-harmonic map flow equation is constructed as a limit of the solutions of $p$-harmonic $\left(p>1\right)$ map flow equation, when the initial data is smooth with small total variation under periodic boundary condition.

We describe a finite-dimensional reduction method to find solutions for a class of slightly supercritical elliptic problems. A suitable truncation argument allows us to work in the usual Sobolev space even in the presence of supercritical nonlinearities: we modify the supercritical term in such a way to have subcritical approximating problems; for these problems, the finite-dimensional reduction can be obtained applying the methods already developed in the subcritical case; finally, we show that, if the truncation is realized at a sufficiently large level, then the solutions of the approximating problems, given by these methods, also solve the supercritical problems when the parameter is small enough.

We study $\left(h\right)$-minimal configurations in Aubry-Mather theory, where $h$ belongs to a complete metric space of functions. Such minimal configurations have definite rotation number. We establish the existence of a set of functions, which is a countable intersection of open everywhere dense subsets of the space and such that for each element $h$ of this set and each rational number $\alpha$, the following properties hold: (i) there exist three different $\left(h\right)$-minimal configurations with rotation number $\alpha$; (ii) any $\left(h\right)$-minimal configuration with rotation number $\alpha$ is a translation of one of these configurations.