| Contents |
| We present some existence and multiplicity results for the following second order periodic system with a nonsmooth potential
\begin{equation}\label{problem}
\left\{ \begin{array}{cc}
-x''(t)-A(t)x(t)\in\partial j(t,x(t)) & \hbox {a.e. on }\, T=[0,b],\\
x(0)=x(b),\ x'(0)=x'(b)\,. & \\
\end{array}
\right .
\end{equation}
Here $A:T\to R^{N\times N}$ is a continuous map and for every $t\in T$, $A(t)$ is a symmetric $N\times N$-matrix. Also $j:T\times R^N\to R$ is a measurable function, which is locally Lipschitz and in general nonsmooth in the $x\in R^N$ variable.
We provide different sets of verifiable hypotheses on $j(t,x)$ ensuring the existence of at least one or two nontrivial solutions of the problem above. In particular, in an existence theorem the Euler functional is coercive and bounded below, in others it is unbounded and in still others it is bounded below but not coercive. Furthermore, in some cases, the analytical framework
incorporates strongly resonant periodic systems. |
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