On the existence of positive solutions of nonlinear second order differential equations
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- by Wei-Cheng Lian, Fu-Hsiang Wong and Cheh-Chih Yeh PDF
- Proc. Amer. Math. Soc. 124 (1996), 1117-1126 Request permission
Abstract:
Under suitable conditions on $f(t,u)$, the boundary value problem \begin{equation*} \begin {cases} (E)~~ u''(t)+f(t,u(t))=0~~~~~\mathrm {in}~~~~~(0,1),\ (BC) \begin {cases} \alpha u(0)-\beta u’(0)=0,\ \gamma u(1)+\delta u’(1)=0\end{cases} \end{cases} \tag *{(BVP)} \end{equation*} has at least one positive solution. Moreover, we also apply this main result to establish several existence theorems of multiple positive solutions for some nonlinear (elliptic) differential equations.References
- C. Bandle, C. V. Coffman, and M. Marcus, Nonlinear elliptic problems in annular domains, J. Differential Equations 69 (1987), no. 3, 322–345. MR 903391, DOI 10.1016/0022-0396(87)90123-9
- C. Bandle and Man Kam Kwong, Semilinear elliptic problems in annular domains, Z. Angew. Math. Phys. 40 (1989), no. 2, 245–257 (English, with French and German summaries). MR 990630, DOI 10.1007/BF00945001
- C. V. Coffman and M. Marcus, Existence and uniqueness results for semi-linear Dirichlet problems in annuli, Arch. Rational Mech. Anal. 108 (1989), no. 4, 293–307. MR 1013459, DOI 10.1007/BF01041066
- H. Dang and K. Schmitt, Existence of positive solutions for semilinear elliptic equations in annular domains, Differential Integral Equations 7 (1994), no. 3-4, 747–758. MR 1270101
- Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
- Lynn Erbe, Boundary value problems for ordinary differential equations, Rocky Mountain J. Math. 1 (1971), no. 4, 709–729. MR 287072, DOI 10.1216/RMJ-1971-1-4-709
- L. H. Erbe, Shou Chuan Hu, and Haiyan Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. 184 (1994), no. 3, 640–648. MR 1281534, DOI 10.1006/jmaa.1994.1227
- L. H. Erbe and Haiyan Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), no. 3, 743–748. MR 1204373, DOI 10.1090/S0002-9939-1994-1204373-9
- Xabier Garaizar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 70 (1987), no. 1, 69–92. MR 904816, DOI 10.1016/0022-0396(87)90169-0
- G. B. Gustafson and K. Schmitt, Nonzero solutions of boundary value problems for second order ordinary and delay-differential equations, J. Differential Equations 12 (1972), 129–147. MR 346234, DOI 10.1016/0022-0396(72)90009-5
- Georges Iffland, Positive solution of a problem of Emden-Fowler type with a free boundary, SIAM J. Math. Anal. 18 (1987), no. 2, 283–292. MR 876271, DOI 10.1137/0518022
- M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 0181881
- Jairo Santanilla, Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations, J. Math. Anal. Appl. 126 (1987), no. 2, 397–408. MR 900756, DOI 10.1016/0022-247X(87)90049-7
- Haiyan Wang, On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Differential Equations 109 (1994), no. 1, 1–7. MR 1272398, DOI 10.1006/jdeq.1994.1042
- Fu Hsiang Wong, Existence of positive solutions of singular boundary value problems, Nonlinear Anal. 21 (1993), no. 5, 397–406. MR 1237131, DOI 10.1016/0362-546X(93)90083-5
Additional Information
- Wei-Cheng Lian
- Affiliation: Department of Mathematics, National Central University, Chung-Li, 32054 Taiwan, Republic of China
- Fu-Hsiang Wong
- Affiliation: Department of Mathematics and Science, National Taipei Teacher’s College, 134, Ho Ping e. Rd. Sec. 2, Taipei 10659, Taiwan, Republic of China
- Cheh-Chih Yeh
- Affiliation: Department of Mathematics, National Central University, Chung-Li, 32054 Taiwan, Republic of China
- Email: Yeh@wangwei.math.ncu.edu.tw
- Received by editor(s): September 21, 1994
- Communicated by: Hal L. Smith
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1117-1126
- MSC (1991): Primary 34B15
- DOI: https://doi.org/10.1090/S0002-9939-96-03403-X
- MathSciNet review: 1328358
Dedicated: Dedicated to Professor Taro Yoshizawa on his $75$th birthday