Consider the following equation $$\Delta v + p(r)e^v = 0$$ on $\mathbb{R}^n$ where $p(r)$ is a polynomial in $r = |(x_1,..., x_n)|$. I want to understand when equations like these have unique solutions, or if they have unique solutions under certain other constraints. I understand the question is very vague and open-ended, but I am looking to basically understand the uniqueness theory for elliptic equations with exponential type nonlinearity. Someone told me that such equations arise in "combustion theory". Any comments/references would be highly appreciated.
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