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The existence of positive solutions for a nonlinear four-point singular boundary value problem with a $$p$$-Laplacian operator. (English) Zbl 1126.34017
The authors study the existence of positive solutions for the following nonlinear four-point singular boundary value problem with a $$p$$-Laplacian operator $(\phi_p(u'))'+a(t)f(u(t))=0,\quad 0<t<1,$
$\alpha\phi_p(u(0))-\beta\phi_p(u'(\xi))=0,\quad \gamma\phi_p(u(1))+\delta\phi_p(u'(\eta))=0,$ where $$\phi_p(s)$$ is a $$p$$-Laplacian operator, i.e. $$\phi_p(s)=| s| ^{p-2}s, p>1, \phi_q=(\phi_p)^{-1}, \frac{1}{p}+\frac{1}{q}=1, \alpha>0, \beta\geq 0, \gamma>0, \delta\geq 0, \xi, \eta\in(0,1)$$ is prescribed and $$\xi<\eta$$, $$a\in C((0,1),(0,\infty))$$. Based on discussions about the asymptotic behavior of the quotient of $$\frac{f(u)}{u^{p-1}}$$ at zero and infinity, and by using the fixed-point index theory, the existence of a positive solution and multiple positive solutions for the above boundary value problem are obtained.

##### MSC:
 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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##### References:
  Wang, H.Y., On the existence of positive solutions for semilinear elliptic equations in the annulus, J. differential equations, 109, 1-7, (1994) · Zbl 0798.34030  Bandle, C.V.; Kwong, M.K., Semilinear elliptic problems in annular domains, J. appl. math. phys. ZAMP, 40, 245-257, (1989) · Zbl 0687.35036  Wei, Z.L., Positive solutions of singular boundary value problems of negative exponent emden – fowler equations, Acts math. sin., 41, 3, 653-662, (1998), (in Chinese) · Zbl 1027.34024  Wei, Z.L., Positive solutions of singular Dirichlet boundary value problems, Chinese ann. math., 20(A), 543-552, (1999), (in Chinese) · Zbl 0948.34501  Gatica, J.A.; Oliker, V.; Waltman, P., Singular boundary value problems for second order ordinary differential equation, J. differential equations, 79, 62-78, (1989) · Zbl 0685.34017  Ma, Y.Y., Positive solutions of singular second order boundary value problems, Acta. math. sin., 41, 6, 1225-1230, (1998), (in Chinese) · Zbl 1027.34025  Kaufmann, E.R.; Kosmatov, N., A multiplicity result for a boundary value problem with infinitely many singularities, J. math. anal. appl., 269, 444-453, (2002) · Zbl 1011.34012  Wong, F.H., The existence of positive solutions for $$m$$-Laplacian BVPs, Appl. math. lett., 12, 12-17, (1999)  He, X.M., The existence of positive solutions of $$p$$-Laplacian equation, Acta math. sin., 46, 4, 805-810, (2003) · Zbl 1056.34033  Liu, B., Positive solutions three-points boundary value problems for one-dimensional $$p$$-Laplacian with infinitely many singularities, Appl. math. lett., 17, 655-661, (2004) · Zbl 1060.34006  Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cone, (1988), Academic Press Sandiego · Zbl 0661.47045  Guo, D.; Lakshmikantham, V.; Liu, X., Nonlinear integral equations in abstract spaces, (1996), Kluwer Academic Publishers · Zbl 0866.45004  Deimling, K., Nonlinear functional analysis, (1980), Springer-Verlag Berlin
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