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

*(English)*Zbl 1065.34021This paper considers the nonlinear Sturm-Liouville problem
\[
-[p(x)u'(x)]' + q(x) u(x) = f(x, u(x)) + h(x), x \in (0, \pi),
\]
under the boundary conditions
\[
c_{00} u(0) + c_{01} u'(0) = 0, c_{10} u(\pi) + c_{11}u'(\pi) = 0,
\]
where \(p \in C^1[0, \pi], q \in C^0 [0, \pi]\) with \(p > 0\) for all \(x \in [0, \pi]\); \(c_{i0}^2 + c_{i1}^2 > 0, i = 0, 1; h \in L^2 (0, \pi)\). It is assumed that \(f\) is continuous and there exist increasing functions \(\zeta_l, \zeta_u:[0, \infty) \rightarrow \mathbb R\), and positive constants \(A, B\) such that \(\lim_{t \rightarrow \infty} \zeta_l (t) = \infty\) and \(-A + \zeta_l (\xi) \xi \leq f(x, \xi) \leq A + \zeta_u (\xi) \xi, \xi \geq 0\) as well as \(| f(x, \xi) | \leq A + B | \xi |, \xi \leq 0\). Thus, the nonlinearity of the equation is superlinear as \(u(x) \rightarrow \infty\) and linearly bounded as \(u(x) \rightarrow - \infty\), so it is said to be asymmetric.

On the other hand, let \(E\) denote the set of \(u \in C^1 [0, \pi]\) satisfying the same boundary conditions described above. \(H = H^2 (0, \pi) \cap E\). The operator \(L\) is defined by \(L u = -(pu')' + q u\) for \(u \in H\). Suppose \((a, b) \in L^\infty (0, \pi)^2\) and let \(\sum_H (a, b)\) denote the set of \(\lambda \in \mathbb R\) for which the equation \(Lu = au^+ - bu^- + \lambda u\) has a nontrivial solution \(u \in H\). Such a \(\lambda\) is called a half-eigenvalue while \(u\) is called a half-eigenfunction. When \(f\) is linearly bounded as \(| \xi |\) tends to \(\infty\), the relation between the sign of certain half-eigenvalues and the existence of the solution has been investigated by the author in [J. Differ. Equations 161, 87–109 (2000; Zbl 0976.34024); ibid. 170, 215–227 (2001; Zbl 0986.34021)].

In this paper, conditions for the existence and nonexistence of the solutions are obtained and are expressed in terms of the signs of the asymptotes of the half-eigenvalues, where these asymptotes are obtained by letting the linear bound on \(f\) tend to \(\infty\) as \(\xi\) tends to \(\infty\).

On the other hand, let \(E\) denote the set of \(u \in C^1 [0, \pi]\) satisfying the same boundary conditions described above. \(H = H^2 (0, \pi) \cap E\). The operator \(L\) is defined by \(L u = -(pu')' + q u\) for \(u \in H\). Suppose \((a, b) \in L^\infty (0, \pi)^2\) and let \(\sum_H (a, b)\) denote the set of \(\lambda \in \mathbb R\) for which the equation \(Lu = au^+ - bu^- + \lambda u\) has a nontrivial solution \(u \in H\). Such a \(\lambda\) is called a half-eigenvalue while \(u\) is called a half-eigenfunction. When \(f\) is linearly bounded as \(| \xi |\) tends to \(\infty\), the relation between the sign of certain half-eigenvalues and the existence of the solution has been investigated by the author in [J. Differ. Equations 161, 87–109 (2000; Zbl 0976.34024); ibid. 170, 215–227 (2001; Zbl 0986.34021)].

In this paper, conditions for the existence and nonexistence of the solutions are obtained and are expressed in terms of the signs of the asymptotes of the half-eigenvalues, where these asymptotes are obtained by letting the linear bound on \(f\) tend to \(\infty\) as \(\xi\) tends to \(\infty\).

Reviewer: Chie-Ping Chu (Taipei)

##### MSC:

34B24 | Sturm-Liouville theory |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

PDF
BibTeX
XML
Cite

\textit{B. P. Rynne}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 57, No. 7--8, 905--916 (2004; Zbl 1065.34021)

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. zeitschrift, 219, 499-513, (1995) · Zbl 0834.35048 |

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

[3] | 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 |

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

[5] | Dancer, E.N., On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. roy. soc. edin., 76A, 283-300, (1977) · Zbl 0351.35037 |

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

[7] | Figueiredo, D.G.; Ruf, B., On a superlinear sturm – liouville equation and a related bouncing problem, J. reine angew. math., 421, 1-22, (1991) · Zbl 0732.34024 |

[8] | Figueiredo, D.G.; Ruf, B., On the periodic fučı́k spectrum and a superlinear sturm – liouville equation, Proc. roy. soc. edin., 123, 95-108, (1993) · Zbl 0813.34029 |

[9] | Hastings, S.P., Boundary value problems in one differential equation with a discontinuity, J. differential equations, 1, 346-369, (1965) · Zbl 0142.06303 |

[10] | 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 |

[11] | Qian, D., Periodic solutions of Liénard equations with superlinear asymmetric nonlinearities, Nonlinear anal. TMA, 43, 637-654, (2001) · Zbl 1047.34052 |

[12] | B. Ruf, Remarks on a superlinear Sturm-Liouville equation, Progress in Partial Differential Equations: Elliptic and Parabolic Problems (Pont-à-Mousson 1991), 232-239, Pitman Res. Notes Math. Ser., 266 (1992). |

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

[14] | Rynne, B.P., Non-resonance conditions for semilinear sturm – liouville problems with jumping non-linearities, J. differential equations, 170, 215-227, (2001) · Zbl 0986.34021 |

[15] | B.P. Rynne, Second order Sturm-Liouville problems with asymmetric, superlinear nonlinearities, in Nonlinear Analysis, TMA, to appear. · Zbl 1037.34019 |

[16] | 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.