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

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

SN - 1230-3429

VL - 32

SP - 21

EP - 40

JO - Topological Methods in Nonlinear Analysis

JF - Topological Methods in Nonlinear Analysis

IS - 1

ER -