Nonlinear Elliptic Equations in Black Hole Holography: Existence, Uniqueness, and Asymptotics

We investigate a nonlinear elliptic boundary value problem that arises naturally in the mathematical formulation of black hole holography. The equation under study is a scalar model that captures the interaction between geometry and exponential nonlinearities on conformally compact manifolds. Our main results establish the existence of weak and strong solutions for all values of the coupling parameter, prove uniqueness in the regime of small coupling, and analyze the breakdown of uniqueness through blow-up phenomena as the parameter increases. We further provide a detailed description of the asymptotic behavior of solutions near the conformal boundary, showing that the leading divergence is universal while the subleading term encodes freely prescribable boundary data. From a variational perspective, the problem admits a natural energy functional whose critical points correspond to solutions, and whose structure reflects stability, multiplicity, and phase transitions. These results illustrate the deep interplay between nonlinear partial differential equations, conformal geometry, and holographic dualities, and they point to further applications of geometric analysis in mathematical physics.

کلیدواژه ها:

Nonlinear elliptic equations ، Variational methods ، conformally compact manifolds ، asymptotic expansions ، AdS/CFT correspondence ، black hole holography