Three positive solutions of $N$-dimensional $p$-Laplacian with indefinite weight

Abstract

This paper is concerned with the global behavior of components of positive radial solutions for the quasilinear elliptic problem with indefinite weight \begin{equation*} \begin{aligned} &\text{div}(\varphi_p(\nabla u))+\lambda h(x)f(u)=0, & & \text{in}\ B,\\ &u=0, & & \text{on}\ \partial B, \end{aligned} \end{equation*} where $\varphi_p(s)=|s|^{p-2}s$, $B$ is the unit open ball of $\mathbb{R}^N$ with $N\geq1$, $10$ is a parameter, $f\in C([0, \infty), [0, \infty))$ and $h\in C(\bar{B})$ is a sign-changing function. We manage to determine the intervals of $\lambda$ in which the above problem has one, two or three positive radial solutions by using the directions of a bifurcation.

Keywords

• positive solutions
• $p$-laplacian
• indefinite weight
• bifurcation