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Communications on Pure and Applied Analysis (CPAA)
 

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

Pages: 905 - 923, Volume 7, Issue 4, July 2008      doi:10.3934/cpaa.2008.7.905

 
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Inbo Sim - Department of Mathematics, Pusan National University, Pusan 609-735, South Korea (email)

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.

Received: July 2007;      Revised: February 2008;      Available Online: April 2008.