14.7 The smooth exponential equationA4D09B787D84

Proof. Suppose that $v-w$ has a positive maximum at a point $x$. Then $\omega _ v(x)\leq \omega _ w(x)$, and taking determinants and applying the two assumptions gives $\mathrm{e}^{\lambda v(x)+F(x)}\leq \mathrm{e}^{\lambda w(x)+F(x)}$, which contradicts $v(x)>w(x)$. For two solutions, apply the comparison in both orders. $\square$

Proof. At a minimum of $u$, the equation gives $\lambda u+F\geq 0$; at a maximum it gives $\lambda u+F\leq 0$. These inequalities prove the stated bounds.

Consider the equations

At $t=0$, $u_0=0$ is a solution. The linearization in $u$ is $L_ u-\lambda $, whose kernel is zero by the maximum principle. Its Fredholm index is zero: deform its principal coefficients to those of a fixed Laplacian while keeping the negative zeroth-order term; the latter operator is invertible, and the index is constant along this deformation of elliptic operators. Thus $L_ u-\lambda $ is an isomorphism between the usual Hölder spaces, with bounded inverse by Schauder theory, and the implicit function theorem proves that the set of solvable parameters is open.

The extremum bounds give $\| u_ t\| _\infty \leq \lambda ^{-1}\| F\| _\infty $. The semilinear version of theorem 14.5.5 and corollary 14.5.8, stated in the remark following that corollary, then gives uniform ellipticity and bounds in every $C^ k$ norm. Along any convergent sequence of solvable parameters, a subsequence of the solutions therefore converges smoothly to a strictly positive solution at the limit parameter. This proves closedness, and hence existence at $t=1$. Uniqueness is proposition 14.7.1. $\square$