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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 $$\lambda = \pi^2/(b-a)^2$$.

##### MSC:
 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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##### References:
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