Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth
In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem \[\begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\tex...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AGH Univeristy of Science and Technology Press
2025-07-01
|
| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4524.pdf |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem \[\begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\text{ in }\Omega, \\ u=0 &\text{ on }\mathbb R^N\setminus \Omega, \end{cases}\] where \(\Omega\subset\mathbb{R}^N\) is bounded domain with smooth boundary, \(1\lt p\lt 2\lt 2^*=\frac{2N}{N-2}\), \(N\geq 3\), \(\lambda\gt 0\), \(M\) is a Kirchhoff coefficient and \(\mathcal{L}\) denotes the mixed local and nonlocal operator. The weight function \(f\in L^{\frac{2^*}{2^*-p}}(\Omega)\) is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions. |
|---|---|
| ISSN: | 1232-9274 |