A second-order Hamiltonian system with time recurrence is studied. The
recurrence condition is weaker than almost periodicity. The existence is
proven of an infinite family of solutions homoclinic to zero
whose support is spread out over
the real line.

An elliptic PDE is studied which is a perturbation of an autonomous
equation. The existence of a nontrivial solution is proven
variational methods. The domain of the equation is unbounded, which
imposes a lack of compactness on the variational problem. In addition,
a popular monotonicity condition on the nonlinearity is not assumed. In
an earlier paper with this assumption, a solution was obtained using a
simple application of topological (Brouwer) degree. Here, a more subtle
degree...

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