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Zbl 1190.34028
Rachuunková, Irena; Tomeček, Jan
Homoclinic solutions of singular nonautonomous second-order differential equations.
(English)
[J] Bound. Value Probl. 2009, Article ID 959636, 21 p. (2009). ISSN 1687-2770/e

This article is concerned with the existence of solutions of the singular boundary value problem $$(p(t)u')'=p(t)f(u),\ t\ge0;\quad u'(0)=0, \quad u(+\infty)=L\tag1$$ Here $p$ is $C^1$, positive and increasing, $p(0)=0$ and $\lim_{t\to\infty}\frac{p'(t)}{p(t)}=0$. $f$ is locally Lipschitz and there are $\bar B<0<L$ such that $f>0$ on $[\bar B,0)$, $f<0$ on $(0, L)$, $f(L)=0$ and, setting $F(x)=-\int_0^xf(z)\,dz$ the equality $F(\bar B)=F(L)$ holds. The authors prove that under these conditions, if in addition $0<\liminf_{x\to-\infty}\frac{|x|}{f(x)}<\infty$, problem (1) has a solution with initial value $u(0)<\bar B$. The arguments involve shooting and connectedness.
[Luis Sanchez (Lisboa)]
MSC 2000:
*34B40 Boundary value problems on infinite intervals
34B15 Nonlinear boundary value problems of ODE
34B16 Singular nonlinear boundary value problems

Keywords: singular boundary value problem; homoclinic solution

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