Existence of sign-changing solutions for one-dimensional $p$-Laplacian problems with a singular indefinite weight
Keywords
Singular one-dimensional p-Laplacian, global bifurcation, existence, Lusternik-Schnirelmann theory, Hardy inequalityAbstract
In this paper, we establish a sequence $\{\nu_k^\infty\}$ of eigenvalues for the following eigenvalue problem $$ \cases \varphi_p (u'(t))' + \nu h(t) \varphi_p(u(t)) = 0 &\text{for } t \in (0,1), \\ u(0) = 0=u(1), \endcases $$ where $\varphi_p(x)=|x|^{p-2}x$, $ 1< p< 2$, $\nu$ a real parameter. In particular, $h \in C((0,1),(0,\infty))$ is singular at the boundaries which may not be of $L^1(0,1)$. Employing global bifurcation theory and approximation technique, we prove several existence results of sign-changing solutions for problems of the form $$ \cases \varphi_p (u'(t))' + \lambda h(t) f (u(t)) = 0 &\text{for } t \in (0,1), \\ u(0) = 0= u(1), \endcases \tag{QP$_\lambda$} $$ when $f \in C({\mathbb{R}}, {\mathbb{R}})$ and $uf(u) > 0$, for all $u \neq 0$ and is odd with various combinations of growth conditions at $0$ and $\infty$.Downloads
Published
2010-04-23
How to Cite
1.
LEE, Yong-Hoon and SIM, Inbo. Existence of sign-changing solutions for one-dimensional $p$-Laplacian problems with a singular indefinite weight. Topological Methods in Nonlinear Analysis. Online. 23 April 2010. Vol. 36, no. 1, pp. 61 - 90. [Accessed 25 September 2024].
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