| Summary: | In this article, we establish a unilateral global bifurcation theorem
from infinity for a class of $N$-dimensional p-Laplacian problems.
As an application, we study the global behavior
of the components of nodal solutions of the problem
$$\displaylines{
\operatorname{div}(\varphi_p(\nabla u))+\lambda a(x)f(u)=0,\quad x\in B,\\
u=0,\quad x\in\partial B,
}$$
where $1<p<\infty$, $\varphi_p(s)=|s|^{p-2}s$,
$B=\{x\in \mathbb{R}^N: |x|<1\}$, and $a\in C(\bar{B}, [0,\infty))$
is radially symmetric with $a\not\equiv 0$ on any subset of
$\bar{B}$, $f\in C(\mathbb{R}, \mathbb{R})$ and there exist two constants
$s_2<0<s_1$, such that $f(s_2)=f(s_1)=0$, and $f(s)s>0$ for
$s\in \mathbb{R}\setminus\{s_2, 0,s_1\}$.
Moreover, we give intervals for the parameter $\lambda$, where
the problem has multiple nodal solutions if
$\lim_{s\to 0}f(s)/\varphi_p(s)=f_0>0$ and
$\lim_{s\to \infty}f(s)/\varphi_p(s)=f_\infty>0$.
We use topological methods and nonlinear
analysis techniques to prove our main results.
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