Nonlinear eigenvalue problems on a half-line and some related topics

Wei-Chuan Wang; Wei-Chuan Wang

doi:10.3934/era.2026372

Electronic Research Archive

2026, Volume 34, Issue 12: 8837-8849. doi: 10.3934/era.2026372

Previous Article Next Article

Research article

Nonlinear eigenvalue problems on a half-line and some related topics

Wei-Chuan Wang ^,

Fundamental General Education Center, National Chin-Yi University of Technology, Taichung 411030, Taiwan

Received: 19 June 2026 Revised: 19 September 2026 Accepted: 30 September 2026 Published: 23 October 2026

The nonlinear eigenvalue problem
$ -\left(|u'(x)|^{p-1}sgn\; u'(x)\right)'+(p-1)\omega(x)|u(x)|^{q-1}sgn \; u(x) = (p-1)\lambda |u(x)|^{p-1}sgn \; u(x), \; \; \; \; 0\leq x< \infty, $
for fixed $ p > q > 1 $, was studied under some conditions on the positive potential $ \omega $. We used various Prüfer angle methods to characterize the eigenvalue properties for problems on the half-line and finite intervals.
- nonlinear Sturm-Liouville equation,
- Prüfer angle method
Citation: Wei-Chuan Wang. Nonlinear eigenvalue problems on a half-line and some related topics[J]. Electronic Research Archive, 2026, 34(12): 8837-8849. doi: 10.3934/era.2026372

Related Papers:

Abstract

The nonlinear eigenvalue problem

$ -\left(|u'(x)|^{p-1}sgn\; u'(x)\right)'+(p-1)\omega(x)|u(x)|^{q-1}sgn \; u(x) = (p-1)\lambda |u(x)|^{p-1}sgn \; u(x), \; \; \; \; 0\leq x< \infty, $

for fixed $ p > q > 1 $, was studied under some conditions on the positive potential $ \omega $. We used various Prüfer angle methods to characterize the eigenvalue properties for problems on the half-line and finite intervals.

References

[1]	P. Binding, P. Drabek, Sturm-Liouville theory for the $p$-Laplacian, Stud. Sci. Math. Hung., 40 (2003), 373–396. https://doi.org/10.1556/sscmath.40.2003.4.1 doi: 10.1556/sscmath.40.2003.4.1
[2]	A. Elbert, A half-linear second order differential equation, in Qualitative Theory of Differential Equations, North-Holland, Amsterdam-New York, (1981), 153–180.
[3]	Y. Huang, G. Metzen, The existence of solutions to a class of semilinear differential equations, Differ. Integr. Equations, 8 (1995), 429–452. https://doi.org/10.57262/die/1369083479 doi: 10.57262/die/1369083479
[4]	N. Dunford, J. T. Schwartz, Linear Operators. Part Ⅱ: Spectral Theory, Self-Adjoint Operators in Hilbert Space, Wiley-Interscience, New York, 1963.
[5]	R. E. Crandall, M. Reno, Ground state energy bounds for potentials $\\|x\\|^\nu$, J. Math. Phys., 23 (1982), 64–70. https://doi.org/10.1063/1.525207 doi: 10.1063/1.525207
[6]	M. Brown, W. Reichel, Eigenvalues of the radially symmetric p-Laplacian, J. Lond. Math. Soc., 69 (2004), 657–675. https://doi.org/10.1112/S002461070300512X doi: 10.1112/S002461070300512X
[7]	M. Brown, M. Eastham, Eigenvalues of the radial $p$-Laplacian with a potential on $(0, \infty)$, J. Comput. Appl. Math., 208 (2006), 111–119. https://doi.org/10.1016/j.cam.2006.10.046 doi: 10.1016/j.cam.2006.10.046
[8]	P. Binding, L. Boulton, P. J. Browne, A Prüfer angle approach to singular Sturm-CLiouville problems with Molčanov potentials, J. Comput. Appl. Math., 208 (2007), 226–234. https://doi.org/10.1016/j.cam.2006.10.032 doi: 10.1016/j.cam.2006.10.032
[9]	A. Molčanov, Conditions for the discreteness of the spectrum of self-adjoint secondorder differential equations, Trudy Mosk. Mat. Obshc., 2 (1953), 169–200.
[10]	P. Binding, P. J. Browne, Generalized Prüfer angle and variational methods for $p$-Laplacian eigenvalue problems on the half-line, Proc. Edinb. Math. Soc., 51 (2008), 565–579. https://doi.org/10.1017/S001309150600157X doi: 10.1017/S001309150600157X
[11]	I. Brinck, Self-adjointness and spectra of Sturm-CLiouville operators, Math. Scand., 7 (1959), 219–239. https://doi.org/10.7146/math.scand.a-10575 doi: 10.7146/math.scand.a-10575
[12]	P. A. Binding, P. J. Browne, B. A. Watson, Weighted $p$-Laplacian problems on a half-line, J. Differ. Equations, 260 (2016), 1372–1391. https://doi.org/10.1016/j.jde.2015.09.045 doi: 10.1016/j.jde.2015.09.045
[13]	K. Kamo, H. Usami, Asymptotic forms of weakly increasing positive solutions for quasilinear ordinary differential equations, Electron. J. Differ. Equations, 126 (2007), 1–12.
[14]	K. Kamo, H. Usami, Characterization of slowly decaying positive solutions of second-order quasilinear ordinary differential equations with subhomogeneity, Bull. Lond. Math. Soc., 42 (2010), 420–428. https://doi.org/10.1112/blms/bdq004 doi: 10.1112/blms/bdq004
[15]	T. Kusano, M. Naito, On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter, Trans. Am. Math. Soc., 354 (2002), 4751–4767. https://doi.org/10.1090/S0002-9947-02-03079-9 doi: 10.1090/S0002-9947-02-03079-9
[16]	M. Naito, On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations, J. Math. Anal. Appl., 381 (2011), 315–327. https://doi.org/10.1016/j.jmaa.2011.04.006 doi: 10.1016/j.jmaa.2011.04.006
[17]	P. Binding, P. J. Browne, I. M. Karabash, Sturm-Liouville problems for the $p$-Laplacian on a half-line, Proc. Edinb. Math. Soc., 53 (2010), 271–291. https://doi.org/10.1017/S001309150800103X doi: 10.1017/S001309150800103X
[18]	M. Brown, M. Eastham, Bounds for the positive eigenvalues of the $p$-Laplacian with decaying potential, J. Math. Anal. Appl., 325 (2007), 734–744. https://doi.org/10.1016/j.jmaa.2006.01.058 doi: 10.1016/j.jmaa.2006.01.058
[19]	P. Drabek, R. Manasevich, On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian, Differ. Integr. Equations, 12 (1999), 773–788. https://doi.org/10.57262/die/1367241475 doi: 10.57262/die/1367241475
[20]	Y. H. Cheng, C. K. Law, W. C. Lian, W. C. Wang, An inverse nodal problem and Ambarzumyan problem for the periodic $p$-Laplacian operator with integrable potentials, Taiwan. J. Math., 19 (2015), 1305–1316. https://doi.org/10.11650/tjm.19.2015.5481 doi: 10.11650/tjm.19.2015.5481
[21]	C. K. Law, W. C. Lian, W. C. Wang, The inverse nodal problem and the Ambarzumyan problem for p-Laplacian, Proc. R. Soc. Edinburgh, Sect. A, 139 (2009), 1261–1273. https://doi.org/10.1017/S0308210508000851 doi: 10.1017/S0308210508000851
[22]	M. Kitano, T. Kusano, On a class second order quasilinear ordinary differential equations, Hiroshima Math. J., 25 (1995), 321–355. https://doi.org/10.32917/hmj/1206127714 doi: 10.32917/hmj/1206127714
[23]	Q. Kong, X. Wang, Nonlinear initial value problem with $p$-Laplacian, Dyn. Syst. Appl., 19 (2010), 33–44.
[24]	Y. Naito, S. Tanaka, Sharp conditions for the existence of sign-changing solutions to the equations involving one-dimensional $p$-Laplacian, Nonlinear Anal., 69 (2008), 3070–3083. https://doi.org/10.1016/j.na.2007.09.002 doi: 10.1016/j.na.2007.09.002

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)