A variant of compactness designated as statistical has been introduced. Statistically compact subsets n-dimensional Euclidean space Rn have characterized by using Cantor-Bendixson derivative and it is further extended to the product RN.

The Palais-Smale Condition C holds for the Yang-Mills functional on principal bundles over compact manifolds of dimension ≤ 3. This was established by S. Sedlacek [17] and C. Taubes [18] Proposition 4.5 using the compactness theorem of K. Uhlenbeck [20]; see also [23]. It is well known that Condition C fails for Yang-Mills over compact manifolds of dimension ≥ 4. The example of SO(3)-invariant ...

We establish an existence result for semilinear elliptic problems with the associated functional not satisfying the Palais-Smale condition. The nonlinearity of our problem does not satisfy the Ambrosetti-Rabinowitz condition.