High school of sciences and technology
Sousse University
4011 Hammam Sousse, Tunisia
email: mohamedlouchaiech@gmail.com
Mohamed Louchaich
Abstract:
This article investigates the fractional singular Kirchhoff system
$$\displaylines{
m(\mathcal{N}(\phi,\psi))\mathcal{L}_\mathcal{K}^p(\phi)
=\lambda a(x)\phi^{-\gamma_1}+\frac{\theta_1}{p^\star_{N,s}}\phi^{\theta_1-1}
\psi^{\theta_2} \quad \text{in } \mathcal{D} \cr
m(\mathcal{N}(\phi,\psi))\mathcal{L}_\mathcal{K}^p(\psi)
=\lambda b(x)\psi^{-\gamma_2}+\frac{\theta_2}{p^\star_{N,s}}\phi^{\theta_1}
\psi^{\theta_2-1} \quad \text{in } \mathcal{D} \cr
\phi>0,\quad \psi>0 \quad\text{in } \mathcal{D} \cr
\phi=\psi=0 \quad \text{in } \mathbb{R}^N\setminus \mathcal{D},
}$$
where
$$
\mathcal{N}(\phi,\psi)=\int_{\mathbb{R}^{2N}}|\phi(x)-\phi(y)|^p \mathcal{K}(x,y)\,dx\,dy
+\int_{\mathbb{R}^{2N}}|\psi(x)-\psi(y)|^p\mathcal{K}(x,y)\,dx\,dy.
$$
Here, \(\mathcal{D}\) is a bounded domain in \(\mathbb{R}^N\) with a
Lipschitz boundary \(\partial\mathcal{D}\).
\(\mathcal{L}_\mathcal{K}^p\) is a non-local operator with
a singular kernel \(\mathcal{K}\).
\(p> 1\), \(\lambda>0\), and \(m\) is a continuous function.
\(\gamma_1,\gamma_2\in (0,1)\). and \(a,b\) are non-negative bounded functions.
\(\theta_1,\theta_2>1\) and \(\theta_1+\theta_2=p^\star_{N_s}\),
where \(p^\star_{N_s}\) is the fractional critical Sobolev exponent
\(p^\star_{N_s}=\frac{Np}{N-sp}\).
Our findings encompass the degenerate case in the fractional
setting, allowing the Kirchhoff function \(m\) to take zero value at zero.
We employ Kajikiya's version of the symmetric mountain pass lemma to prove the
existence of a sequence of infinitely many small solutions with negative energy
that converge to zero.
Submitted September 20, 2025. Published January 17, 2026.
Math Subject Classifications: 35A15, 35D30, 35J60, 35J75, 35R11, 46E35, 47G20.
Key Words: p-Laplacian; Kirchhoff system; critical point theory; multiple positive solution;
singular nonlinearity, critical Sobolev exponent.
DOI: 10.58997/ejde.2026.05
Show me the PDF file (434 KB), TEX file for this article.
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Mohamed Louchaich High school of sciences and technology Sousse University 4011 Hammam Sousse, Tunisia email: mohamedlouchaiech@gmail.com |
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