Opuscula Math. 43, no. 1 (2023), 47-66

Opuscula Mathematica

Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball

John R. Graef
Doudja Hebboul
Toufik Moussaoui

Abstract. In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the \(p\)-Laplacian \[-\Big(a+b\int_{\Omega_e}|\nabla u|^p dx\Big)\Delta_p u=\lambda f\left(|x|,u\right),\ x\in \Omega_e,\quad u=0\ \text{on} \ \partial\Omega_e,\] where \(\lambda \gt 0\) is a parameter, \(\Omega_e = \lbrace x\in\mathbb{R}^N : |x|\gt r_0\rbrace\), \(r_0\gt 0\), \(N \gt p \gt 1\), \(\Delta_p\) is the \(p\)-Laplacian operator, and \(f\in C(\left[ r_0, +\infty\right)\times\left[0,+\infty\right),\mathbb{R})\) is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of \(\lambda\).

Keywords: Kirchhoff problem, \(p\)-Laplacian, positive radial solution, variational methods.

Mathematics Subject Classification: 35A01, 35A15, 35B38, 35D30, 35J92.

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  • Doudja Hebboul
  • Ecole Normale Supérieure, Laboratory of Partial Differential Equations and History of Mathematics, Kouba, Algiers, Algeria
  • Toufik Moussaoui
  • Ecole Normale Supérieure, Laboratory of Fixed Point Theory and Applications, Kouba, Algiers, Algeria
  • Communicated by Vicentiu D. Rădulescu.
  • Received: 2022-08-17.
  • Revised: 2022-11-07.
  • Accepted: 2022-11-08.
  • Published online: 2022-12-30.
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Cite this article as:
John R. Graef, Doudja Hebboul, Toufik Moussaoui, Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball, Opuscula Math. 43, no. 1 (2023), 47-66, https://doi.org/10.7494/OpMath.2023.43.1.47

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