We consider the singular Liouville equation and the Henon-Lane-Emden problem on simply connected planar domains. We show that any solution to each problem must satisfy a uniform bound on the mass. The same results applies to some systems and more general non-linearities. The proofs are based on the Riemann mapping theorem and a Pohozaev-type identity.
An interpolation approach to $L^{\infty}$ a priori estimates for elliptic problems with nonlinearity on the boundary
Regularity versus singularities for elliptic problems in two dimensions
Topics in elliptic problems: from semilinear equations to shape optimization
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