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

Galanis, George N. ; Palamides, Alex P.

Electronic journal of differential equations, Tome 2005 (2005)

Résumé

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 01 $

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 $u=0$.

Détail

Classification : 34B15, 34B18
Keywords: three-point singular boundary-value problem, p-Laplacian, positive and negative solutions, vector field, nonlinear alternative of Leray-Schauder