SOME RECENT RESULTS ON THE SPECTRUM OF MULTI-POINT EIGENVALUE PROBLEMS FOR THE p-LAPLACIAN

TitleSOME RECENT RESULTS ON THE SPECTRUM OF MULTI-POINT EIGENVALUE PROBLEMS FOR THE p-LAPLACIAN
Publication TypeJournal Article
Year of Publication2011
AuthorsGENOUD, FRANCOIS, RYNNE, BRYANP
Volume15
Issue4
Start Page413
Pagination22
Date Published2011
ISSN1083-2564
AMS34B08, 34B10, 34L05, 47J15, 47N20
Abstract

In this paper we describe some recent results regarding the spectral properties of p-Laplacian problems subject to various forms of multi-point boundary conditions. In particular, we consider Dirichlet and Neumann-type boundary conditions, and mixtures of these conditions. We also consider certain types of nonlocal boundary conditions expressed in terms of Stieltjes integrals, which have been discussed recently and which generalize the previously considered multi-point conditions. It is shown that, under suitable assumptions, the structure of the spectrum and the properties of the eigenfunctions for these boundary value problems with nonlocal boundary conditions are very similar to those of the classical linear Sturm-Liouville problem with separated boundary conditions. Results on global bifurcation, non-resonance conditions and existence of nodal solutions for related nonlinear problems are also presented. In a final section we discuss the necessity of some of the hypotheses we impose, and outline some open problems.

URLhttp://www.acadsol.eu/en/articles/15/4/1.pdf
Refereed DesignationRefereed
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REFERENCES
[1] P. Binding, P. Dr´abek, Sturm-Liouville theory for the p-Laplacian, Studia Sci. Math. Hungar.
40 (2003), 375–396.
[2] C. Bai, J. Fang, Existence of multiple positive solutions for nonlinear m-point boundary value
problems, J. Math. Anal. Appl. 281 (2003), 76–85.
[3] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New
York, 1955.
[4] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[5] N. Dodds, B. P. Rynne, Spectral properties and nodal solutions for second-order, m-point,
p-Laplacian boundary value problems, Topol. Methods Nonlinear Anal. 32 (2008), 21–40.
[6] J. Gao, D. Sun, M. Zhang, Structure of Eigenvalues of Multi-point Boundary Value Problems,
preprint.
[7] M. Garc´ıa-Huidobro, Ch. P. Gupta, R. Man´asevich, Some multipoint boundary value problems
of Neumann-Dirichlet type involving a multipoint p-Laplace like operator, J. Math. Anal. Appl.
333 (2007), 247–264.
[8] M. Garc´ıa-Huidobro, R. Man´asevich, J. R. Ward, A homotopy along p for systems with a
p-Laplace operator, Adv. Differential Equations 8 (2003), 337–356.
[9] C. P. Gupta, A non-resonant generalized multi-point boundary-value problem of Dirichelet
type involving a p-Laplacian type operator, in: Proceedings of the Sixth Mississippi State–UBA
Conference on Differential Equations and Computational Simulations, 127–139, Electron. J.
Differ. Equ. Conf., 15, Southwest Texas State Univ., San Marcos, TX, 2007.
[10] E. Hewitt, K. Stromberg, Real and Abstract Analysis, Second Edition, Springer, 1969.
[11] Y. X. Huang, G. Metzen, The existence of solutions to a class of semilinear differential equations,
Differential Integral Equations 8 (1995), 429–452.
[12] A. N. Kolmogorov, S. V. Fomin, Introductory Real Analysis, Dover, New York, 1975.
[13] P. Lindqvist, Some remarkable sine and cosine functions, Ricerche di Matematica 44 (1995),
269–290.
[14] Y. Liu, Non-homogeneous boundary-value problems of higher order differential equations with
p-Laplacian, Electron. J. Differential Equations 2008, No. 22.
[15] R. Ma, D. O’Regan, Nodal solutions for second-order m-point boundary value problems with
nonlinearities across several eigenvalues, Nonlinear Anal. 64 (2006), 1562–1577.
[16] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Analysis 7
(1971), 487-513
[17] B. P. Rynne, Spectral properties and nodal solutions for second-order, m-point, boundary
value problems, Nonlinear Analysis 67 (2007), 3318–3327.
[18] B. P. Rynne, Spectral properties of second-order, multi-point, p-Laplacian boundary value
problems, Nonlinear Analysis 72 (2010), 4244–4253.
[19] B. P. Rynne, Spectral properties of p-Laplacian problems with Neumann and mixed-type
multi-point boundary conditions, Nonlinear Analysis 74 (2010), 1471–1484.
[20] B. P. Rynne, Eigenvalue criteria for existence of positive solutions of second-order, multi-point,
p-Laplacian boundary value problems, Topol. Methods Nonlinear Anal. 36 (2010), 311–326.
[21] J. R. L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems: a unified
approach, J. London Math. Soc. 74 (2006), 673–693. 
[22] J. R. L. Webb, K. Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of
nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal.
27 (2006), 91–115.
[23] X. Xu, Multiple sign-changing solutions for some m-point boundary-value problems, Electron.
J. Differential Equations 89 (2004).
[24] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. I - Fixed Point Theorems,
Springer-Verlag, New York, 1986.