Geodesic
Note on the uniqueness of a global positive solution to the second Painlevé equation
Electronic journal of differential equations, Tome 2001 (2001)
The purpose of this note is to study the uniqueness of solutions to $ u'' -u^3 + (t-c)u = 0$, for $ t \in (0,+\infty)$ with Neumann condition at 0. Assuming a certain conditon at infinity, Helfer and Weissler [6] have found a unique solution. We show that, without any assumptions at infinity, this problem has exactly one global positive solution. Moreover, the solution behaves like $\sqrt{t}$ as $t$ approaches infinity.
Classification : 34B15, 35B05, 82D55
Keywords: second Painlevé equation, Neumann condition, global existence 
@article{EJDE_2001__2001__a169,
     author = {Guedda,  Mohammed},
     title = {Note on the uniqueness of a global positive solution to the second {Painlev\'e} equation},
     journal = {Electronic journal of differential equations},
     year = {2001},
     volume = {2001},
     zbl = {0983.34082},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a169/}
}
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Guedda,  Mohammed. Note on the uniqueness of a global positive solution to the second Painlevé equation. Electronic journal of differential equations, Tome 2001 (2001). http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a169/