School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: wjchen@swu.edu.cn
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: ycdaimath@163.com
Wenjing Chen, Yunchuan Dai
Abstract:
We study the mixed local and nonlocal Kirchhoff problem
$$\displaylines{
M\Big(\int_{\Omega}|\nabla u|^p\,dx+
\iint_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dx\,dy\Big)
\bigl(-\Delta_pu+(-\Delta)^s_pu\bigr)
=f(x,u)+\lambda|u|^{p^*-2}u, \quad
\text{in }\Omega,\cr
u=0,\quad \text{in }\mathbb R^N\setminus\Omega,
}$$
where \(\Omega\subset\mathbb R^N\) is a bounded smooth domain,
\(N\geq 3\), \(0< s< 1< p< N\), \(p^*=Np/(N-p)\) and \(\lambda\geq 0\).
\(M\) is a continuous function that vanishes at the origin.
\(f\) is a Caratheodory function that is odd in its second variable.
When \(M(t)\) is comparable to \(t^{\alpha-1}\) with \(1< \alpha< p^*/p\),
Krasnosel'skii genus theory gives infinitely many pairs of nontrivial weak
solutions for \(\lambda=0\) under subcritical growth assumptions on \(f\).
If the lower-order growth exponent exceeds \(p\), we combine a
truncation with Moser iteration. For each positive integer \(k\), we obtain at least
\(k\) distinct pairs of nontrivial weak solutions when \(\lambda\) is sufficiently small,
with the upper bound on \(\lambda\) depending on \(k\).
In the Kirchhoff critical case \(\alpha=p^*/p\), we prove the Palais-Smale
condition and obtain infinitely many pairs of nontrivial weak solutions for
\(0\leq \lambda< AS^{p^*/p}\), where \(A\) is the lower bound coefficient of \(M\)
and \(S\) is the mixed Sobolev constant.
Submitted December 13, 2025. Published October 11, 2026.
Math Subject Classifications: 35A15, 35J60, 35R11.
Key Words: Local and nonlocal operators; Kirchhoff problems; multiple solutions; genus theory.
DOI: 10.58997/ejde.2026.80
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Wenjing Chen School of Mathematics and Statistics Southwest University Chongqing 400715, China email: wjchen@swu.edu.cn |
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Yunchuan Dai School of Mathematics and Statistics Southwest University Chongqing 400715, China email: ycdaimath@163.com |
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