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Homoclinic solutions of singular nonautonomous second-order differential equations. (English) Zbl 1190.34028
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.

34B40Boundary value problems for ODE on infinite intervals
34B15Nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
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