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Second-order Sturm–Liouville problems with asymmetric, superlinear nonlinearities. (English) Zbl 1037.34019
The author considers the nonlinear Sturm-Liouville problem \[ -(p(x)u'(x))'+q(x)u(x)=f(x,u(x),u'(x)), \text{ in }(0,\pi), \] \[ c_{00}u(0)+c_{01}u'(0)=0, \quad c_{10}u(\pi)+c_{11}u'(\pi)=0, \] where \(p\in C^1[0,\pi]\), \(q\in C[0,\pi]\), with \(p(x)>0\) for all \(x\in[0,\pi]\), and \(c_{i0}^2+c_{i1}^2>0\), \(i=0,1\). Further, it is supposed that \(f:[0,\pi]\times \mathbb{R}^2\to\mathbb{R}\) is continuous, superlinear as \(u\to\infty\), and linearly bounded as \(u\to-\infty\). The main results are presented in form of two theorems (the second one with a stronger condition on \(f\)) in which conditions are given guaranteeing the existence of solutions to the above problem having special nodal properties. The results generalize those due to A. Capietto and W. Dambrosio [Nonlinear Anal., Theory Methods Appl. 38A, No. 7, 869–896 (1999; Zbl 0952.34012)] in several aspects. Quite detailed comparison between those results is given, as well as a comparison with some results from D. Arcoya and S. Villegas [Math. Z. 219, No.4, 499–513 (1995; Zbl 0834.35048)] and from H. Berestycki [J. Differ. Equations 26, 375–390 (1977; Zbl 0331.34020)].
Reviewer: Pavel Rehak (Brno)

MSC:
34B24 Sturm-Liouville theory
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