Non-existence and uniqueness results for the extended Fisher-Kolmogorov equation

We give a classification of all bounded solutions of the equation [ u'''' + p u'' + F'(u) = 0, qquad -infty < t< infty, ] in which $F$ is a general quartic polynomial and $p$ is restricted to various subsets of $(-infty,0]$. These results are obtained by combining an a priori estimate with geometric arguments in the $(u,u'')$-plane.