Consider the boundary value problem $$ \begin{cases}y''+cy'+f(y)=0, & -M<x<M\\y(-M)=1, y(M)=0, & 0\leq y\leq 1\end{cases}\tag{M,c} $$

where $f$ is some Lipschnitz non-linearity with $f(0)=f(1)=0$.

Use the sub- and super-solution approach to prove that for each pair (M,c) there exists a solution and that this solution is monotonous.

I think the existence is a direct consequence of the fact that $u=0$ is a subsolution and $v=1$ is a supersolution.

But what about the monotonicity of the existing solution? Is this a consequence of the strong maximum principle?


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