$-p_{xx}+f(p)=0$ has a unique solution $p$

Question

In a paper I am reading they have the following as a Lemma without proof. So I am trying to prove it myself.

Suppose that $f:\mathbb{R}\to \mathbb{R}$ that satisfies the following:

1. $f'(0) >0$
2. There exists $\eta$ such that $F(\eta) <0$ where $F'=f$ and $F(0)=0$.
3. If $u_0$ is the zero of $F$ with smallest nonzero absolute value, then $f(u_0) \neq 0$

If $f$ satisfies all of the above then $$-p_{xx}+f(p)=0 $$

Has a unique solution $p(x)$ that satisfies

1. $p(x)>0$, $p(x)=p(-x)$, $p(0)=u_0$
2. $p(x)$ decays exponentially like $e^{-c|x|}$ with $c>0$.

I thought of using the implicit function theorem. But the solution $p(x)$ I am seeking has global properties. So how should I go about solving this problem?. Can anybody give some hints as to how I should proceed?.

EDIT The paper I am refering to is: Grillakis, Shatah and Strauss: Stability theory of solitary waves in the presence of symmetry I, Journal of Functional analysis 74, pp 160-197 (1987).It is very dense reading as you would expect from a research paper. But I am making progress, about half a page per week progress, but progress nonetheless. I am reading only the examples towards the end :). This is one is on page 186 btw.