Existence and multiplicity of solutions for mixed local and nonlocal Kirchhoff type problems
- 1Southwest Univ., Chongqing, China
Abstract
We study the mixed local and nonlocal Kirchhoff problem displaylines M(∫Ω|∇ u|p dx+ iintℝ2N(|u(x)−u(y)|p)/(|x−y|N+ps) dx dy) (−Δpu+(−Δ)spu) =f(x,u)+λ|u|p*−2u, in Ω,cr u=0, in ℝN∖Ω, where Ω⊂ℝN is a bounded smooth domain, N≥ 3, 0< s< 1< p< N, p*=Np/(N−p) and λ≥ 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 tα−1 with 1< alpha< p*/p, Krasnosel'skii genus theory gives infinitely many pairs of nontrivial weak solutions for λ=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 λ is sufficiently small, with the upper bound on λ depending on k. In the Kirchhoff critical case α=p*/p, we prove the Palais-Smale condition and obtain infinitely many pairs of nontrivial weak solutions for 0≤ λ< ASp*/p, where A is the lower bound coefficient of M and S is the mixed Sobolev constant.
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BibTeX
@article{chen2026existence,
title = {{Existence and multiplicity of solutions for mixed local and nonlocal Kirchhoff type problems}},
author = {Wenjing Chen and Yunchuan Dai},
journal = {Electronic Journal of Differential Equations},
year = {2026},
month = oct,
url = {http://ejde.math.txstate.edu/Volumes/2026/80/abstr.html},
note = {CC BY},
}
RIS
TY - JOUR
TI - Existence and multiplicity of solutions for mixed local and nonlocal Kirchhoff type problems
AU - Wenjing Chen
AU - Yunchuan Dai
PY - 2026
DA - 2026/10/23
JO - Electronic Journal of Differential Equations
UR - http://ejde.math.txstate.edu/Volumes/2026/80/abstr.html
AB - We study the mixed local and nonlocal Kirchhoff problem displaylines M(∫_Ω|∇ u|^p dx+ iint_ℝ^2N(|u(x)−u(y)|^p)/(|x−y|^N+ps) dx dy) (−Δ_pu+(−Δ)^s_pu) =f(x,u)+λ|u|^p^*−2u, in Ω,cr u=0, in ℝ^N∖Ω, where Ω⊂ℝ^N is a bounded smooth domain, N≥ 3, 0< s< 1< p< N, p^*=Np/(N−p) and λ≥ 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^α−1 with 1< alpha< p^*/p, Krasnosel'skii genus theory gives infinitely many pairs of nontrivial weak solutions for λ=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 λ is sufficiently small, with the upper bound on λ depending on k. In the Kirchhoff critical case α=p^*/p, we prove the Palais-Smale condition and obtain infinitely many pairs of nontrivial weak solutions for 0≤ λ< AS^p^*/p, where A is the lower bound coefficient of M and S is the mixed Sobolev constant.
SN - 1072-6691
ER -