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

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 $$s\neq0$$.

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
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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
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