Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation

@article{Marino2016ExistenceAA,
  title={Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation},
  author={Greta Marino and Sunra Mosconi},
  journal={arXiv: Classical Analysis and ODEs},
  year={2016}
}
In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to $u""+qu"+f(u)=0$, under various hypotheses on the parameter $q$ and on the potential $F(t)=\int_0^tf(s)\, ds$, generally assumed to be bounded from below. We prove a non-existence result in the case $q\le 0$ and an existence result of periodic solution for: 1) almost every suitably small (depending on $F$), positive values of $q$; 2) all suitably large (depending on $F$) values… 
1 Citations
Existence and asymptotic properties of solutions for a nonlinear Schrödinger elliptic equation from geophysical fluid flows
TLDR
The existence theorem and asymptotic properties of radial positive solutions are established by using a new renormalization technique.

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