Existence results of sign-changing solutions for singular one-dimensional \(p\)-Laplacian problems. (English) Zbl 1138.34010

Lee, Yong-Hoon; Sim, Inbo

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 5, 1195-1209 (2008).

Applying the global bifurcation theorem and deriving the shape of the unbounded subcontinua of solutions, the authors establish existence and multiplicity of sign-changing solutions for the boundary value problem
\[ (\varphi_p(u'(t)))'+h(t)f(u(t))=0,\;t\in(0,1), \]
\[ u(0)=0=u(1), \]
where \(\varphi_p(x)=| x| ^{p-2}x, p>1,\) \(h\) may by singular at \(t=0\) and/or \(t=1\) and \(f\in C(\mathbb R,\mathbb R)\). The cases
\[ f_0:=\lim_{| u| \to0}{f(u)\over \varphi_p(u)}=0\text{ and }f_0=\infty \]
are considered. In the first case the authors suppose in addition that
\[ f_\infty:=\lim_{| u| \to\infty}{f(u)\over \varphi_p(u)}=\infty,\quad h\in L^1(0,1),\;h\geq0\text{ for a.e. }t\in(0,1),\;\int_Ih(s)\,ds>0, \]
where \(I\) is any compact subinterval of \((0,1)\), and \(sf(s)>0\) for \(s\neq0\). Under these assumptions, they prove that for each \(k\in \mathbb N\), the considered problem has two solutions \(u^+_k\) and \(u^-_k\) such that \(u^+_k\) has exactly \(k-1\) zeros and is positive near \(t=0\), and \(u^-_k\) has exactly \(k-1\) zeros and is negative near \(t=0\). A similar result is obtained in the second case under the following assumptions:
\[ f_\infty=0,\quad h\in C^1(0,1)\cap L^1(0,1),\;h\geq0\text{ for a.e. }t\in(0,1), \]
\(\lim_{t\to0^{+}}th(t)\) and \(\lim_{t\to1^{-}}(1-t)h(t)\) exist and \(sf(s)>0\) for \(s\neq0\).

Reviewer: Petio S. Kelevedjiev (Sliven)

MSC:

34B16	Singular nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations

Keywords:

Singular boundary value problem; p-Laplacian; existence; multiplicity; global bifurcation theorem

References:

[1]	Agarwal, R.P.; Lü, H.; O’Regan, D., Eigenvalues and the one-dimensional \(p\)-Laplacian, J. math. anal. appl., 266, 383-400, (2002) · Zbl 1002.34019
[2]	García-Huidobro, M.; Manásevich, R.; Ward, J.R., A homotopy along \(p\) for systems with a vector \(p\)-Laplace operator, Adv. differential equations, 8, 337-356, (2003) · Zbl 1039.34010
[3]	Kong, L.; Wang, J., Multiple positive solutions for the one-dimensional \(p\)-Laplacian, Nonlinear anal., 42, 1327-1333, (2000) · Zbl 0961.34012
[4]	Kusano, T.; Jaros, T.; Yoshida, N., A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order, Nonlinear anal., 40, 381-395, (2000) · Zbl 0954.35018
[5]	Lee, Y.H.; Sim, I., Global bifurcation phenomena for singular one-dimensional \(p\)-Laplacian, J. differential equations, 229, 229-256, (2006) · Zbl 1113.34010
[6]	Ma, R., Nodal solutions second-order two-point boundary value problems with superlinear or sublinear nonlinearities, J. math. anal. appl., (2006), (online available)
[7]	Ma, R.; Thompson, B., Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities, J. math. anal. appl., 303, 726-735, (2005) · Zbl 1075.34017
[8]	Manásevich, R.; Mawhin, J., Periodic solutions of nonlinear systems with \(p\)-Laplacian-like operators, J. differential equations, 145, 367-393, (1998) · Zbl 0910.34051
[9]	Manásevich, R.; Mawhin, J., Boundary value problems for nonlinear perturbations of vector \(p\)-Laplacian-like operators, J. Korean math. soc., 37, 665-685, (2000) · Zbl 0976.34013
[10]	Naito, Y.; Tanaka, S., On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations, Nonlinear anal., 56, 919-935, (2004) · Zbl 1046.34038
[11]	Y. Naito, S. Tanaka, Multiplicity of solutions for a class of two-point boundary value problems involving one-dimensional \(p\)-Laplacian (preprint)
[12]	Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. funct. anal., 7, 487-513, (1971) · Zbl 0212.16504
[13]	Rabinowitz, P.H., Some aspects of nonlinear eigenvalue problems, Rocky mountain J. math., 3, 161-202, (1973) · Zbl 0255.47069
[14]	Sánchez, J., Multiple positive solutions of singular eigenvalue type problems involving the one-dimensional \(p\)-Laplacian, J. math. anal. appl., 292, 401-414, (2004) · Zbl 1057.34012
[15]	K. Schmitt, R. Thompson, Nonlinear analysis and differential equations: An introduction, University of Utah Lecture Note, Salt Lake City, 2004
[16]	Wang, J., The existence of positive solutions for the one-dimensional \(p\)-Laplacian, Proc. amer. math. soc., 125, 2275-2283, (1997) · Zbl 0884.34032
[17]	Yang, X., Sturm type problems for singular \(p\)-Laplacian boundary value problems, Appl. math. comput., 136, 181-193, (2003) · Zbl 1035.34013