Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains

We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\R$ and $C>0$. We provide a complete characterization of the set of $(p,\sigma)\in\R^2$ such that the equation has no positive (super-)solutions, depending on the values of $A,B$ and the principle Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmen--Lindel\"of type comparison arguments and an improved version of Hardy's inequality in cone--like domains.Comment: 30 pages, 1 figur

Modular and duality properties of surface operators in N=2* gauge theories

We calculate the instanton partition function of the four-dimensional N=2* SU(N) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from S-duality, we show that the effective twisted superpotential, which governs the infrared dynamics of the two-dimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. We also show that our results, derived for monodromy defects in the four-dimensional theory, match the effective twisted superpotential describing the infrared properties of certain two-dimensional sigma models coupled either to pure N=2 or to N=2* gauge theories.Comment: 51 pages, v3: references added, typos fixed, footnote added, some small changes in the text, appendix B streamlined. Matches the published versio