Electron. J. Diff. Equ., Vol. 2014 (2014), No. 13, pp. 1-6.

Bifurcation from infinity and nodal solutions of quasilinear elliptic differential equations

Bian-Xia Yang

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
 \hbox{div}(\varphi_p(\nabla u))+\lambda a(x)f(u)=0,\quad x\in B,\cr
 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.

Submitted November 29, 2013. Published January 8, 2014.
Math Subject Classifications: 35P30,35B32.
Key Words: p-Laplacian; bifurcation; nodal solutions.

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Bian-Xia Yang
School of Mathematics and Statistics
Lanzhou University
Lanzhou, Gansu 730000, China
email: yanglina7765309@163.com

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