Second-order Sturm–Liouville problems with asymmetric, superlinear nonlinearities.

*(English)*Zbl 1037.34019The 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 |

PDF
BibTeX
XML
Cite

\textit{B. P. Rynne}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 54, No. 5, 939--947 (2003; Zbl 1037.34019)

Full Text:
DOI

##### References:

[1] | Arcoya, D.; Villegas, S., Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at −∞ and superlinear at +∞, Math. Z., 219, 499-513, (1995) · Zbl 0834.35048 |

[2] | Berestycki, H., On some nonlinear sturm – liouville problems, J. differential equations, 26, 375-390, (1977) · Zbl 0331.34020 |

[3] | Capietto, A.; Dambrosio, W., Multiplicity results for some two-point superlinear asymmetric boundary value problems, Nonlinear anal., 3, 869-896, (1999) · Zbl 0952.34012 |

[4] | Castro, A.; Shivaji, R., Multiple solutions for a Dirichlet problem with jumping nonlinearities. II, J. math. anal. appl., 133, 509-528, (1988) · Zbl 0695.34018 |

[5] | Coddington, E.A.; Levinson, N., Theory of ordinary differential equations, (1955), McGraw-Hill New York · Zbl 0042.32602 |

[6] | Drabek, P.; Invernizzi, S., On the periodic BVP for the forced Duffing equation with jumping nonlinearity, Nonlinear anal., 10, 643-650, (1986) · Zbl 0616.34010 |

[7] | Fabry, C.; Habets, P., Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. math., 60, 266-276, (1993) · Zbl 0779.34019 |

[8] | Perera, K., Existence and multiplicity results for a sturm – liouville equation asymptotically linear at −∞ and superlinear at +∞, Nonlinear anal., 39, 669-684, (2000) · Zbl 0942.34021 |

[9] | Rynne, B.P., The Fucik spectrum of general sturm – liouville problems, J. differential equations, 161, 87-109, (2000) · Zbl 0976.34024 |

[10] | Zinner, B., Multiplicity of solutions for a class of superlinear sturm – liouville problems, J. math. anal. appl., 176, 282-291, (1993) · Zbl 0784.34023 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.