##
**Multiple solutions for a class of nonlinear Sturm-Liouville problems on the half line.**
*(English)*
Zbl 0703.34032

The author considers a class of nonlinear Sturm-Liouville problems. He gives an extensive list of references containing fourty references together with the complete proofs of the following: 7 theorems, 17 remarks, 17 lemmas, 5 propositions and two corollaries which are pieced together to construct essentially the author’s doctoral thesis from the University of Wisconsin Madison (USA) where the reviewer worked as a visiting professor for fourteen months. In the first section the author obtains existence and uniqueness for positive and negative solutions for the problems
\[
(1)\quad -u^ n=\lambda r(x)u-F(x,u)u,\quad 0<x<+\infty,\quad u(a)\cos \theta -u'(a) \sin \theta =0,\quad u\in L^ 2[a,\infty),
\]
where r and F are nonnegative continuous functions, \(a\geq 0\) and \(0\leq \theta \leq \pi /2.\)

In section 2 he obtains the existence of solutions with prescribed number of nodes when the nonlinearitiy is odd near zero. In section 3 the author considers analogous results for radical solutions in higher dimensional class, while in section 4 he considers the partial differential equation \[ (2)\quad \Delta \hat u=\lambda \hat r(x)\hat u(x)-\hat F(x,\hat u(x))\hat u(x),\quad x\in {\mathbb{R}}, \] where \(\hat r:\) \({\mathbb{R}}^ N\to (0,\infty)\) and \(\hat F:\) \({\mathbb{R}}^ N\times {\mathbb{R}}\to (0,\infty)\) are radially symmetric. The author seeks the solution of (2) in the form \(\hat u\in L^ 2({\mathbb{R}}^ N)\cap C^ 2({\mathbb{R}}^ N)\).

In section 2 he obtains the existence of solutions with prescribed number of nodes when the nonlinearitiy is odd near zero. In section 3 the author considers analogous results for radical solutions in higher dimensional class, while in section 4 he considers the partial differential equation \[ (2)\quad \Delta \hat u=\lambda \hat r(x)\hat u(x)-\hat F(x,\hat u(x))\hat u(x),\quad x\in {\mathbb{R}}, \] where \(\hat r:\) \({\mathbb{R}}^ N\to (0,\infty)\) and \(\hat F:\) \({\mathbb{R}}^ N\times {\mathbb{R}}\to (0,\infty)\) are radially symmetric. The author seeks the solution of (2) in the form \(\hat u\in L^ 2({\mathbb{R}}^ N)\cap C^ 2({\mathbb{R}}^ N)\).

Reviewer: F.M.Ragab

### Keywords:

nonlinear Sturm-Liouville problems
PDF
BibTeX
XML
Cite

\textit{C. Chen}, J. Differ. Equations 85, No. 2, 236--275 (1990; Zbl 0703.34032)

Full Text:
DOI

### References:

[1] | Amick, C.J, Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids, Ann. scuola norm. sup. Pisa cl. sci. (4), 11, 441-499, (1984) · Zbl 0568.35076 |

[2] | Benci, V; Fortunato, D, Does bifurcation from the essential spectrum occur?, Comm. partial differential equations, 6, 249-272, (1981) · Zbl 0471.35007 |

[3] | Berestycki, H; Lions, P.-L, Nonlinear scalar field equations, Arch. rational mech. anal., 82, 313-375, (1983) |

[4] | Birkhoff, G; Rota, G, Ordinary differential equations, (1978), Wiley New York · Zbl 0183.35601 |

[5] | Bongers, A; Heinz, H.-P; Küpper, T, Existence and bifurcation theorems for non-linear elliptic eigenvalue problems on unbounded domains, J. differential equations, 47, 327-357, (1983) · Zbl 0506.35081 |

[6] | Chen, C.N, Multiple solutions and bifurcation for a class of nonlinear Sturm-Liouville eigenvalue problems on an unbounded domain, () |

[7] | {\scC. N. Chen}, Uniqueness and bifurcation for solutions of nonlinear Sturm-Liouville eigenvalue problems, Arch. Rational Mech. Anal., in press. · Zbl 0712.34035 |

[8] | {\scC. N. Chen}, Multiple solutions for a class of nonlinear Sturm-Liouville problems when nonlinearities are not odd, J. Differential Equations, in press. · Zbl 0779.34018 |

[9] | Chiapinelli, R; Stuart, C.A, Bifurcation when the linearized problem has no eigenvalues, J. differential equations, 30, 296-307, (1978) · Zbl 0419.34010 |

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

[11] | Courant, R; Hilbert, D, () |

[12] | Crandall, M.G; Rabinowitz, P.H, Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. math. mech., 29, 1083-1102, (1970) · Zbl 0206.09705 |

[13] | Esteban, M.J, Multiple solutions of semilinear elliptic problems in a ball, J. differential equations, 57, 112-137, (1985) · Zbl 0519.35031 |

[14] | Gilbarg, D; Trudinger, N.S, Elliptic partial differential equations of second order, (1977), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0691.35001 |

[15] | Hartman, P, The L^{2}-solution of linear differential equations of second order, Duke math. J., 14, 323-326, (1947) · Zbl 0029.29401 |

[16] | Heinz, H.-P, Nodal properties and variational characterizations of solutions to nonlinear Sturm-Liouville problems, J. differential equations, 62, 299-333, (1986) · Zbl 0579.34011 |

[17] | Heinz, H.-P, Nodal properties and bifurcation from the essential spectrum for a class of nonlinear Sturm-Liouville problems, J. differential equations, 64, 79-108, (1986) · Zbl 0593.34021 |

[18] | Heinz, H.-P, Free Ljusternik-schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems, J. differential equations, 66, 263-300, (1987) · Zbl 0607.34012 |

[19] | Hempel, J.A, Multiple solutions for a class of nonlinear boundary value problems, Indiana univ. math. J., 20, 983-996, (1971) · Zbl 0225.35045 |

[20] | Hempel, J.A, Superlinear variational boundary value problems and nonuniqueness, () |

[21] | Jones, C; Küpper, T, Characterization of bifurcation from the continuous spectrum by nodal properties, J. differential equations, 54, 196-220, (1984) · Zbl 0547.34018 |

[22] | Jones, C; Küpper, T, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. math. anal., 17, No. 4, 803-835, (1986) · Zbl 0606.35032 |

[23] | Küpper, T, The lowest point of the continuous spectrum as a bifurcation point, J. differential equations, 34, 212-217, (1979) · Zbl 0388.47040 |

[24] | Küpper, T, On minimal nonlinearities which permit bifurcation from the continuous spectrum, Math. methods appl. sci., 1, 572-580, (1979) · Zbl 0437.34010 |

[25] | Küpper, T; Riemer, D, Necessary and sufficient conditions for bifurcation from the continuous spectrum, Nonlinear anal. (TMA), 3, 555-561, (1979) · Zbl 0388.47039 |

[26] | Nehari, Z, Characteristic values associated with a class of nonlinear second-order differential equations, Acta math., 105, 141-175, (1961) · Zbl 0099.29104 |

[27] | Protter, M.H; Weinberger, H.F, Maximum principles in differential equations, (1967), Prentice-Hall Englewood Cliffs, NJ · Zbl 0153.13602 |

[28] | Rabinowitz, P.H, A note on a nonlinear eigenvalue problem for a class of differential equations, J. differential equations, 9, 536-548, (1971) · Zbl 0218.34027 |

[29] | Rabinowitz, P.H, Global aspects of bifurcation, () · Zbl 0212.16504 |

[30] | Rabinowitz, P.H, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Comm. pure appl. math., 23, 939-961, (1970) · Zbl 0206.09706 |

[31] | Rabinowitz, P.H, Some global results for nonlinear eigenvalue problems, J. fund. anal., 7, 487-513, (1971) · Zbl 0212.16504 |

[32] | Rabinowitz, P.H, Some aspects of nonlinear eigenvalue problems, Rocky mountain J. math., 3, 161-202, (1973) · Zbl 0255.47069 |

[33] | Rudin, W, Principles of mathematical analysis, (1976), McGraw-Hill New York · Zbl 0148.02903 |

[34] | Ryder, G.H, Boundary value problems for a class of nonlinear differential equations, Pacific J. math., 22, 477-503, (1967) · Zbl 0152.28303 |

[35] | Stuart, C.A, Bifurcation from the essential spectrum, () · Zbl 0888.47045 |

[36] | Stuart, C.A, Bifurcation for Dirichlet problems without eigenvalues, (), 169-192 · Zbl 0505.35010 |

[37] | Stuart, C.A, Bifurcation for Neumann problems without eigenvalues, J. differential equations, 36, 391-407, (1980) · Zbl 0468.34009 |

[38] | Stuart, C.A, Global properties of components of solutions of non-linear second order ordinary differential equations on the half-line, Ann. scuola norm. sup. Pisa cl. sci., 2, 265-286, (1975) · Zbl 0326.34034 |

[39] | Toland, J.F, Global bifurcation for Neumann problems without eigenvalues, J. differential equations, 44, 82-110, (1982) · Zbl 0455.34015 |

[40] | Turner, R.E.L, Superlinear Sturm-Liouville problems, J. differential equations, 13, 157-171, (1973) · Zbl 0272.34031 |

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.