Bifurcation from infinity and nodal solutions of quasilinear elliptic differential equations

Abstract

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 $10$ 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.

Keywords

• p-Laplacian
• bifurcation
• nodal solutions