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Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

2 votes
1 answer
160 views

The Monge- Ampère equation with a non positive right hand side

Let $\Omega$ be a domain, $u$ and $f$ are real valued functions on $\Omega$, $(u_{ij})$ is the Hessian matrix of $u$. The function $f$ may change sign: that said, do there exist solutions for the foll …
liding's user avatar
  • 153
1 vote
0 answers
157 views

Is $(e^{u}+1)\Delta u+u=0$ the Euler-Lagrange equation of a functional energy?

Does there exist a functional energy $I$ such that $$(e^{u}+1)\Delta u+u=0$$ is the Euler-Lagrange equation associated with the energy functional $I$?
liding's user avatar
  • 153