Spectral properties and nodal solutions for second-order, m-point, p-Laplacian boundary value problems

Niall Dodds, Bryan P. Rynne

    Research output: Contribution to journalArticle

    Abstract

    NOTE: THE MATHEMATICAL SYMBOLS IN THIS ABSTRACT CANNOT BE DISPLAYED CORRECTLY ON THIS PAGE. PLEASE REFER TO THE PUBLISHER'S WEBSITE FOR AN ACCURATE DISPLAY. We consider the m-point boundary value problem consisting of the equation ?? p(u0)0 = f(u); on (0; 1); (1) together with the boundary conditions u(0) = 0; u(1) = mX??2 i=1 iu( i); (2) where p > 1, p(s) := jsjp??1sgn s, s 2 R, m 3, i; i 2 (0; 1), for i = 1; : : : ;m ?? 2, and mX??2 i=1 i < 1: We assume that the function f : R ! R is continuous, satis es sf(s) > 0 for s 2 R n f0g, and that f0 := lim !0 f( ) p( ) > 0 (we assume that the limit exists and is nite). Closely related to the problem (1), (2), is the spectral problem consisting of the equation ?? p(u0)0 = p(u); (3) together with the boundary conditions (2). It is shown that the spectral prop- erties of (2), (3), are similar to those of the standard Sturm-Liouville problem with separated (2-point) boundary conditions (with a minor modi cation to deal with the multi-point boundary condition). The topological degree of a related operator is also obtained. These spectral and degree theoretic results are then used to prove a Rabinowitz-type global bifurcation theorem for a bi- furcation problem related to the problem (1), (2). Finally, we use the global bifurcation theorem to obtain nodal solutions (that is, sign-changing solutions with a speci ed number of zeros) of (1), (2), under various conditions on the asymptotic behaviour of f.
    Original languageEnglish
    Pages (from-to)21-40
    Number of pages20
    JournalTopological Methods in Nonlinear Analysis
    Volume32
    Issue number1
    Publication statusPublished - 2008

    Fingerprint

    Nodal Solutions
    P-Laplacian
    Spectral Properties
    Boundary value problems
    Boundary Value Problem
    Boundary conditions
    Global Bifurcation
    M-point Boundary Value Problem
    Sign-changing Solutions
    Topological Degree
    Sturm-Liouville Problem
    Spectral Problem
    Theorem
    Websites
    Minor
    Display
    Bifurcation
    Positive ions
    Asymptotic Behavior
    Display devices

    Cite this

    @article{74f551135e1b43ef9cad5f182aea4554,
    title = "Spectral properties and nodal solutions for second-order, m-point, p-Laplacian boundary value problems",
    abstract = "NOTE: THE MATHEMATICAL SYMBOLS IN THIS ABSTRACT CANNOT BE DISPLAYED CORRECTLY ON THIS PAGE. PLEASE REFER TO THE PUBLISHER'S WEBSITE FOR AN ACCURATE DISPLAY. We consider the m-point boundary value problem consisting of the equation ?? p(u0)0 = f(u); on (0; 1); (1) together with the boundary conditions u(0) = 0; u(1) = mX??2 i=1 iu( i); (2) where p > 1, p(s) := jsjp??1sgn s, s 2 R, m 3, i; i 2 (0; 1), for i = 1; : : : ;m ?? 2, and mX??2 i=1 i < 1: We assume that the function f : R ! R is continuous, satis es sf(s) > 0 for s 2 R n f0g, and that f0 := lim !0 f( ) p( ) > 0 (we assume that the limit exists and is nite). Closely related to the problem (1), (2), is the spectral problem consisting of the equation ?? p(u0)0 = p(u); (3) together with the boundary conditions (2). It is shown that the spectral prop- erties of (2), (3), are similar to those of the standard Sturm-Liouville problem with separated (2-point) boundary conditions (with a minor modi cation to deal with the multi-point boundary condition). The topological degree of a related operator is also obtained. These spectral and degree theoretic results are then used to prove a Rabinowitz-type global bifurcation theorem for a bi- furcation problem related to the problem (1), (2). Finally, we use the global bifurcation theorem to obtain nodal solutions (that is, sign-changing solutions with a speci ed number of zeros) of (1), (2), under various conditions on the asymptotic behaviour of f.",
    author = "Niall Dodds and Rynne, {Bryan P.}",
    note = "dc.publisher: Juliusz Schauder Center for Nonlinear Studies",
    year = "2008",
    language = "English",
    volume = "32",
    pages = "21--40",
    journal = "Topological Methods in Nonlinear Analysis",
    issn = "1230-3429",
    publisher = "Juliusz Schauder Center",
    number = "1",

    }

    Spectral properties and nodal solutions for second-order, m-point, p-Laplacian boundary value problems. / Dodds, Niall; Rynne, Bryan P.

    In: Topological Methods in Nonlinear Analysis, Vol. 32, No. 1, 2008, p. 21-40.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Spectral properties and nodal solutions for second-order, m-point, p-Laplacian boundary value problems

    AU - Dodds, Niall

    AU - Rynne, Bryan P.

    N1 - dc.publisher: Juliusz Schauder Center for Nonlinear Studies

    PY - 2008

    Y1 - 2008

    N2 - NOTE: THE MATHEMATICAL SYMBOLS IN THIS ABSTRACT CANNOT BE DISPLAYED CORRECTLY ON THIS PAGE. PLEASE REFER TO THE PUBLISHER'S WEBSITE FOR AN ACCURATE DISPLAY. We consider the m-point boundary value problem consisting of the equation ?? p(u0)0 = f(u); on (0; 1); (1) together with the boundary conditions u(0) = 0; u(1) = mX??2 i=1 iu( i); (2) where p > 1, p(s) := jsjp??1sgn s, s 2 R, m 3, i; i 2 (0; 1), for i = 1; : : : ;m ?? 2, and mX??2 i=1 i < 1: We assume that the function f : R ! R is continuous, satis es sf(s) > 0 for s 2 R n f0g, and that f0 := lim !0 f( ) p( ) > 0 (we assume that the limit exists and is nite). Closely related to the problem (1), (2), is the spectral problem consisting of the equation ?? p(u0)0 = p(u); (3) together with the boundary conditions (2). It is shown that the spectral prop- erties of (2), (3), are similar to those of the standard Sturm-Liouville problem with separated (2-point) boundary conditions (with a minor modi cation to deal with the multi-point boundary condition). The topological degree of a related operator is also obtained. These spectral and degree theoretic results are then used to prove a Rabinowitz-type global bifurcation theorem for a bi- furcation problem related to the problem (1), (2). Finally, we use the global bifurcation theorem to obtain nodal solutions (that is, sign-changing solutions with a speci ed number of zeros) of (1), (2), under various conditions on the asymptotic behaviour of f.

    AB - NOTE: THE MATHEMATICAL SYMBOLS IN THIS ABSTRACT CANNOT BE DISPLAYED CORRECTLY ON THIS PAGE. PLEASE REFER TO THE PUBLISHER'S WEBSITE FOR AN ACCURATE DISPLAY. We consider the m-point boundary value problem consisting of the equation ?? p(u0)0 = f(u); on (0; 1); (1) together with the boundary conditions u(0) = 0; u(1) = mX??2 i=1 iu( i); (2) where p > 1, p(s) := jsjp??1sgn s, s 2 R, m 3, i; i 2 (0; 1), for i = 1; : : : ;m ?? 2, and mX??2 i=1 i < 1: We assume that the function f : R ! R is continuous, satis es sf(s) > 0 for s 2 R n f0g, and that f0 := lim !0 f( ) p( ) > 0 (we assume that the limit exists and is nite). Closely related to the problem (1), (2), is the spectral problem consisting of the equation ?? p(u0)0 = p(u); (3) together with the boundary conditions (2). It is shown that the spectral prop- erties of (2), (3), are similar to those of the standard Sturm-Liouville problem with separated (2-point) boundary conditions (with a minor modi cation to deal with the multi-point boundary condition). The topological degree of a related operator is also obtained. These spectral and degree theoretic results are then used to prove a Rabinowitz-type global bifurcation theorem for a bi- furcation problem related to the problem (1), (2). Finally, we use the global bifurcation theorem to obtain nodal solutions (that is, sign-changing solutions with a speci ed number of zeros) of (1), (2), under various conditions on the asymptotic behaviour of f.

    M3 - Article

    VL - 32

    SP - 21

    EP - 40

    JO - Topological Methods in Nonlinear Analysis

    JF - Topological Methods in Nonlinear Analysis

    SN - 1230-3429

    IS - 1

    ER -