Electron. J. Diff. Eqns., Vol. 2005(2005), No. 106, pp. 1-18.

Positive solutions of three-point boundary-value problems for p-Laplacian singular differential equations

George N. Galanis, Alex P. Palamides

Abstract:
In this paper we prove the existence of positive solutions for the three-point singular boundary-value problem
$$ -[\phi _{p}(u')]'=q(t)f(t,u(t)),\quad 0<t<1 $$
subject to
$$ u(0)-g(u'(0))=0,\quad u(1)-\beta u(\eta )=0 $$
or to
$$ u(0)-\alpha u(\eta )=0,\quad u(1)+g(u'(1))=0, $$
where $\phi _{p}$ is the $p$-Laplacian operator, $0<\eta <1$; $0<\alpha ,\beta <1$ are fixed points and $g$ is a monotone continuous function defined on the real line $\mathbb{R}$ with $g(0)=0$ and $ug(u)\geq 0$. Our approach is a combination of Nonlinear Alternative of Leray-Schauder with the properties of the associated vector field at the $(u,u')$ plane. More precisely, we show that the solutions of the above boundary-value problem remains away from the origin for the case where the nonlinearity is sublinear and so we avoid its singularity at $u=0$.

Submitted May 13, 2005. Published October 7, 2005.
Math Subject Classifications: 34B15, 34B18.
Key Words: Three-point singular boundary-value problem; p-Laplacian; positive and negative solutions; vector field; Nonlinear alternative of Leray-Schauder.

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George N. Galanis
Naval Academy of Greece
Piraeus, 185 39, Greece
email: ggalanis@math.uoa.gr
Alex P. Palamides
Department of Communication Sciences
University of Peloponnese
22100 Tripolis, Greece
email: palamid@uop.gr

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