Three positive solutions of $N$-dimensional $p$-Laplacian with indefinite weight - Tianlan Chen; Ruyun Ma (Electronic Journal of Qualitative Theory of Differential Equations, 2019)
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 $N\geq1$, $1<p<\infty$, $\lambda>0$ 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.