Using a new manner to rescale fields in N = 2 gauged supergravity with n V vector multiplets and n H hypermultiplets, we develop the explicit derivation of the rigid limit of quaternionic isometry Ward identities agreeing with known results. We show that the rigid limit can be achieved, amongst others, by performing two successive transformations on the covariantly holomorphic sections V M (z,z) of the special Kahler manifold: a particular symplectic change followed by a particular Kahler transformation. We also give a geometric interpretation of the η i parameters used in arXiv:1508.01474 to deal with the expansion of the holomorphic prepotential F (z) of the N = 2 theory. We give as well a D-brane realisation of gauged quaternionic isometries and an interpretation of the embedding tensor ϑ u M in terms of type IIA/IIB mirror symmetry. Moreover, we construct explicit metrics for a new family of 4r-dimensional quaternionic manifolds M (n H ) QK classified by ADE Lie algebras generalising the SO (1, 4) /SO (4) geometry which corresponds to A 1 ∼ su (2). The conditions of the partial breaking of N = 2 supersymmetry in the rigid limit are also derived for both the observable and the hidden sectors. Other features are also studied.

We consider the U (1) n extension of the effective N = 2 supersymmetric U (1)× U (1) model of arXiv:1204.2141; and study the explicit relationship between partial breaking of N = 2 supersymmetry constraint and D3 brane tadpole anomaly of type IIB string on Calabi-Yau threefolds in presence of H RR and H N S fluxes. We also comment on supersymmetry breaking in the particular N = 2 U (1) Maxwell theory; and study its interpretation in connection with the tadpole anomaly with extra localized flux sources.Key words: Coulomb branch of N = 2 QFT 4 's, Partial breaking of N = 2 supersymmetry, Tadpole anomaly of type IIB on CY3, Brane realisation.

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In this paper, we prove the existence of infinitely many solutions of a system of boundary value problems involving flux boundary conditions in anisotropic variable exponent Sobolev spaces, by applying a critical point variational principle obtained by Ricceri as a consequence of a more general variational principle and the theory of the anisotropic variable exponent Sobolev spaces.

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