On the existence of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition

Inbo Sim

doi:10.3934/cpaa.2008.7.905

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2008, Volume 7, Issue 4: 905-923. Doi: 10.3934/cpaa.2008.7.905

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On the existence of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition

Inbo Sim^1,

1.
Department of Mathematics, Pusan National University, Pusan 609-735

Received: June 30, 2007

Revised: January 31, 2008

Published: June 2008

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Abstract

We obtain the existence results of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition:

$\varphi (u'(t))' + \lambda h(t) f (u(t)) = 0,\ \ $ a.e. $\ t \in (0,1), \qquad\qquad\qquad\qquad\qquad $ $(\Phi_\lambda)$

$u(0) = 0=u(1),$

where $\varphi : \mathbb R \to \mathbb R$ is an odd increasing homeomorphism, $\lambda$ a positive parameter and $h \in L^1(0,1)$ a nonnegative measurable function on $(0,1)$ which may be singular at $t = 0$ and/or $t = 1,$ and $f \in C(\mathbb R, \mathbb R)$ and is odd.

Keywords:

Singular one-dimensional $\varphi$-Laplacian,
global bifurcation,
existence,
asymptotic condition.

Mathematics Subject Classification: Primary: 34A37; Secondary: 34B15.

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