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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
Full Text:
References:
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