Attainability at the borderline threshold p=2q

Determine whether the global infimum of the energy functional is attained at the threshold p=2q for the resonant (p,q)-Laplacian and indefinite double-phase problems with q>1 and κ=1/2.

Background

Theorem \ref{thm:geometry} establishes boundedness and negativity of the infimum in the borderline case pκ=q but does not establish whether the infimum is attained. For the resonant (p,q)-Laplacian and indefinite double-phase problems discussed earlier in the paper, the relevant parameters satisfy q>1 and κ=1/2, so the unresolved borderline condition is p=2q. Resolving attainability at this threshold would determine whether the variational infimum is realized by a nonzero critical point in these model problems.

References

For the problems eq:intro-models:1 and eq:intro-models:2 considered in , one has $q>1$ and $\kappa=1/2$ (see Section~\ref{sec:double-phase} below), and the attainability question remains open at the threshold $p=2q$.

— On the generalized Fredholm alternative for the $p$-Laplacian with resonant subhomogeneous terms  (2609.25768 - Bobkov, 22 Sep 2026) in Remark \ref{rem:borderline}, Section \ref{sec:geometry}