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

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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About the authors ▼
John R. Graef (corresponding author)
https://orcid.org/0000-0002-8149-4633
University of Tennessee at Chattanooga, Department of Mathematics, Chattanooga, TN 37403, USA
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
About the article ▼
Communicated by Vicentiu D. Rădulescu.
Received: 2022-08-17.
Revised: 2022-11-07.
Accepted: 2022-11-08.
Published online: 2022-12-30.
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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