Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs):
\textbf{1. The Yamabe Type Equation (for $n>2$):}
- The Yamabe Type Equation (for $n>2$):
\begin{equation} -\Delta u + hu = fu^{2^*-1}, \end{equation}
where $2^*$$2^*=\frac{2n}{n-2}$ is the critical exponent associated with the Sobolev embedding $W^{1,2} \subset L^{p}$.
\textbf{2. The Liouville Equation (for $n=2$):}
- The Liouville Equation (for $n=2$):
\begin{equation} -\Delta u + h = f e^{2u}. \end{equation}
In the case of the Liouville equation ($n=2$), it is also considered critical, with $2^* = +\infty$. However, it's worth noting that in this case, the role of the Sobolev embedding is played by the Moser-Trudinger inequality.
These equations naturally arise in conformal geometry, particularly in the context of the evolution of the scalar curvature (or Gaussian curvature for $n=2$) under a conformal change of metric. While these equations have found significant applications in differential geometry, I believe their universal property of conformal invariance suggests potential applications beyond this field.
I am currently preparing a talk for a diverse audience, including individuals from mathematical economics. I would greatly appreciate any references or examples of concrete fields of application for these equations, as their relevance may extend beyond their prominent role in differential geometry and conformal field theory.
Thank you in advance for your input and suggestions.
