Global Palais–Smale classification for the local p-Laplacian on unbounded domains

Establish a full Struwe-type global compactness decomposition for Palais–Smale sequences of critical local p-Laplacian problems on unbounded domains, including the classification of sequences without a sign condition when p\ne2.

Background

The paper places its fractional Hardy–Sobolev global compactness theorem in the broader context of Struwe-type decompositions for critical elliptic equations. Although complete decompositions are available in certain bounded-domain settings, the nonlinear p-Laplacian lacks the Hilbert-space orthogonality used in the semilinear case, making profile extraction and energy decoupling substantially more difficult.

The authors explicitly identify the unresolved issue as the general classification of Palais–Smale sequences for the local p-Laplacian on unbounded domains, particularly in the absence of a sign condition. Resolving it would extend global compactness theory to settings where concentration can occur both in the domain and at infinity.

References

Without a sign condition on the (PS) sequence, even the bounded-domain decomposition of is not known to hold in full generality, and the corresponding classification on unbounded domains for $p\ne2$ remains, to a large extent, open.

— Critical fractional $p$-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions  (2609.08510 - Biswas et al., 8 Sep 2026) in Section 1, Literature review

An analogous global compactness decomposition exhibiting both the pure Sobolev and the Hardy-Sobolev profiles at $\al=0$ is, to the best of our knowledge, not known even for the corresponding local $p$-Laplace operator (with $p\neq2$) on any unbounded domain.

— Critical fractional $p$-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions  (2609.08510 - Biswas et al., 8 Sep 2026) in Section 1, Literature review