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Some surprising results on a one-dimensional elliptic boundary value blow-up problem. (English) Zbl 0945.34007
The paper is concerned with the existence of sign-changing solutions of the boundary blow-up problem \[ \Delta_{p}u=\lambda f(u), \quad u(a)=u(b)=+\infty, \tag{*} \] where \(\lambda\) is a positive parameter, \(f\geq 0\) and \(\Delta_{p}u\) is the 1-dimensional \(p\)-Laplacian on \((a,b)\) (i.e., \(\Delta_{p}u = (|u'(t)|^{p-2}u'(t))'\), \(a<t<b\)).
The author studies, for some classes of functions \(f\), how the number of sign-changing solutions varies with \(\lambda\). A sample result: Let \(p=2\) (i.e., \(\Delta_{p}u=u''\)), \(f(u)=\alpha u^q\) for \(u\geq 0\) and \(f(u) = (1+\beta\sin|u|)|u|\) for \(u\leq 0\), with \(q>3\) and \(0<\beta<1\). Then there exist \(\lambda_{1}\), \(\lambda_{2}\) such that (*) has at least one sign-changing solution if \(\lambda\geq\lambda_{1}\), exactly one sign-changing solution if \(\lambda>\lambda_{2}\) and no sign-changing solutions if \(\lambda<\lambda_{1}\). Furthermore, for each \(n\geq 1\) there is \(\delta>0\) such that (*) has at least \(n\) sign-changing solutions for \(\lambda\in (\pi^2/(b-a)^2-\delta, \pi^2/(b-a)^2+\delta)\). In particular, there are infinitely many such solutions if \(\lambda = \pi^2/(b-a)^2\).

34B15 Nonlinear boundary value problems for ordinary differential equations
35J70 Degenerate elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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