Opuscula Math. 44, no. 1 (2024), 5-17
https://doi.org/10.7494/OpMath.2024.44.1.5
Opuscula Mathematica
Uniqueness for a class p-Laplacian problems when a parameter is large
Abstract. We prove uniqueness of positive solutions for the problem \[-\Delta_{p}u=\lambda f(u)\text{ in }\Omega,\ u=0\text{ on }\partial \Omega,\] where \(1\lt p\lt 2\) and \(p\) is close to 2, \(\Omega\) is bounded domain in \(\mathbb{R}^{n}\) with smooth boundary \(\partial \Omega\), \(f:[0,\infty)\rightarrow [0,\infty )\) with \(f(z)\sim z^{\beta }\) at \(\infty\) for some \(\beta \in (0,1)\), and \(\lambda\) is a large parameter. The monotonicity assumption on \(f\) is not required even for \(u\) large.
Keywords: singular \(p\)-Laplacian, uniqueness, positive solutions.
Mathematics Subject Classification: 35J92, 35J75.
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- B. Alreshidi
- Mississippi State University, Department of Mathematics and Statistics, Mississippi State, MS 39762, USA
- D.D. Hai (corresponding author)
- Mississippi State University, Department of Mathematics and Statistics, Mississippi State, MS 39762, USA
- Communicated by Vicenţiu D. Rădulescu.
- Received: 2023-02-22.
- Revised: 2023-08-23.
- Accepted: 2023-08-27.
- Published online: 2023-10-27.
