Homoclinic solutions of singular nonautonomous second-order differential equations
This article is concerned with the existence of solutions of the singular boundary value problem \[ (p(t)u')'=p(t)f(u),\;t\geq0;\quad u'(0)=0, \quad u(+\infty)=L\tag{1} \] 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.
scientific article; zbMATH DE number 5626180