Theorem 10.31.45DDDCE55695E1

Proof. For each integer $j\geq 1$, the preceding theorem gives a solution of

Put $S=\sup _ Xu_1$ and $w_ j=u_1+(j-1)S$. Then $w_ j/j\geq u_1$, and hence

Exponential order at exponent $1/j$ gives $u_ j\leq w_ j$. Define

The upper bound just proved and the positive lower volume $\ell $ imply

The envelope identity gives $v_ j\leq \varphi $. Theorem 5.25.2 gives a subsequence $j_ k$ along which $v_{j_ k}\to u$ in $L^1$, where $u\not\equiv -\infty $ and $\sup _ Xu=0$, and $c_{j_ k}\to c>0$. Fixed-dominator continuity gives $u\in \mathcal E_\varphi $ and $\mu _{v_{j_ k}}\rightharpoonup \mu _ u$.

Lemma 5.44.5, which holds for every smooth background, gives $\mathrm{e}^{v_ j/j}\to 1$ in $L^1(\mu )$. Consequently $c_{j_ k}\mathrm{e}^{v_{j_ k}/j_ k}\mu \to c\mu $ in total variation, and the limiting equation is $\mu _ u=c\mu $. Integrating it and using (10.31.1) gives the stated bounds for $c$. The scalar comparison in corollary 13.19.3 gives uniqueness of $c$. $\square$