We classify deformations of [Formula: see text]-module structure on the space [Formula: see text], where [Formula: see text] is the space of [Formula: see text]-densities on [Formula: see text].

In this work, we describe the H-invariant, so(n)-relative cohomology of a natural class of osp(n|2)-modules M, for n ≠ 2. The Lie superalgebra osp(n|2) can be realized as a superalgebra of vector fields on the superline R1|n. This yields canonical actions on spaces of densities and differential operators on the superline. The above result gives the zero, first, and second cohomology spaces for these modules of densities and differential operators.

We establish some existence results of polyharmonic boundary value problems with supercritical growth. Our approach is based on truncation argument as well as L ∞ -bounds. Also, by virtue of Pucci-serrin's variational identity [19], we improve previous non-existence results.

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