Existence of sign-changing solutions for one-dimensional $p$-Laplacian problems with a singular indefinite weight

Yong-Hoon Lee, Inbo Sim

Abstract


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 $\infty$.

Keywords


Singular one-dimensional p-Laplacian; global bifurcation; existence; Lusternik-Schnirelmann theory; Hardy inequality

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