POSITIVE RADIAL SOLUTIONS FOR A SEMIPOSITONE PROBLEM OF ELLIPTIC KIRCHHOFF EQUATIONS WITH SUBLINEAR NONLINEARITIES

Tingting Zhang; Ruyun Ma; Meng Yan

doi:10.11948/20250061

2025 Volume 15 Issue 6

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Tingting Zhang, Ruyun Ma, Meng Yan. POSITIVE RADIAL SOLUTIONS FOR A SEMIPOSITONE PROBLEM OF ELLIPTIC KIRCHHOFF EQUATIONS WITH SUBLINEAR NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3769-3781. doi: 10.11948/20250061

Citation:

Tingting Zhang, Ruyun Ma, Meng Yan. POSITIVE RADIAL SOLUTIONS FOR A SEMIPOSITONE PROBLEM OF ELLIPTIC KIRCHHOFF EQUATIONS WITH SUBLINEAR NONLINEARITIES[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3769-3781. doi: 10.11948/20250061

POSITIVE RADIAL SOLUTIONS FOR A SEMIPOSITONE PROBLEM OF ELLIPTIC KIRCHHOFF EQUATIONS WITH SUBLINEAR NONLINEARITIES

1.
School of Mathematics and Statistics, Xidian University, Xi'an, Shanxi 710071, China
2.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Author Bio: Email: zhangtingting2215@163.com(T. Zhang); Email: yanmeng161999@163.com(M. Yan)
Corresponding author: Ruyun Ma, ryma@xidian.edu.cn(R. Ma)
Fund Project: The authors were supported by National Natural Science Foundation of China(12461036)

Abstract

Abstract

We study the semipositone problem of the elliptic Kirchhoff type equation
$ \begin{eqnarray} \left\{ \begin{array}{cc} \begin{aligned} &-\Big(b\int_{\Omega_e}|\nabla u|^2dx\Big)\Delta u=\lambda K(|x|)f(u), \ \ &x\in B_e,\\ &u(x)=0, \ \ &|x|=r_0,\\ &u(x)\rightarrow0, \ \ &|x|\rightarrow \infty, \end{aligned} \end{array} \right. \end{eqnarray} $
where$ b $is a positive constant,$ \lambda $is a positive parameter,$ B_e=\{x\in\mathbb{R}^N:\, |x|>r_0\} $,$ N>2 $,$ K : [r_0, +\infty) \to (0, +\infty) $is continuous with$ r^{N+\eta} K(r) $bounded for some$ \eta > 0 $,$ f : [0, +\infty) \to \mathbb{R} $is continuous,$ f(0)<0 $and$ \underset{u\to\infty}\lim\frac{f(u)}{u^q} =\beta $for some$ q\in (0, 1] $. We show that there exists$ \lambda^*>0 $, such that (0.1) has at least one positive radial solution if$ \lambda>\lambda^* $. The proof of the main result is based upon bifurcation theory.
- Kirchhoff equation /
- semipositone problem /
- positive radial solution /
- bifurcation
MSC: 34B10, 34B15, 34B18, 34C23, 35B09

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References

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