Three positive nonconstant radial solutions of nonlinear Neumann problems with indefinite weight

Abstract

In this article, we consider the global behavior of components of positive nonconstant radial solutions for the Neumann problem with indefinite weight $\left\{\begin{array}{ll}-\mathrm{\Delta }u=\lambda h\left(x\right)f\left(u\right),& \mathrm{i}\mathrm{n}B,\\ {\mathrm{\partial }}_{\nu }u=0,& \mathrm{o}\mathrm{n}\mathrm{\partial }B,\end{array}\right.$where $\lambda >0$ is a parameter, $B\subseteq {\mathbb{R}}^N(N\ge 1)$ is the unit open ball, $f\in C\left(\right[0,\mathrm{\infty }),[0,\mathrm{\infty }\left)\right)$ and $h\in C\left(\overline{B}\right)$ satisfying ${\int }_{B}h\left(x\right)\mathrm{d}x<0$ is the sign-changing function. We determine the intervals of λ in which the above problem has one, two or three positive nonconstant radial solutions by using the directions of a bifurcation.

Keywords:

• Positive nonconstant solutions
• Neumann boundary conditions
• indefinite weight
• bifurcation

2020 Mathematics Subject Classifications:

• 34B18
• 35J60

Acknowledgments

This work was supported by the NSFC (No. 11801453, No. 12061064, No. 11901464), Gansu Provincial National Science Foundation of China (No. 20JR10RA100).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the NSFC [grant numbers 11801453, 12061064, 11901464], Gansu Provincial National Science Foundation of China [grant number 20JR10RA100].