Kirchhoff-type equations involving the Fractional $(p,q)-$Laplacian
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912549009620992 |
|---|---|
| author | Carrero, Lisbeth Hernández-Llanos, Pedro |
| author_facet | Carrero, Lisbeth Hernández-Llanos, Pedro |
| contents | In this paper, we study the existence and nonexistence of solutions for the following Kirchhoff-type fractional $(p\text{-}q)$-Laplacian problem:
\begin{equation*} \begin{cases}
M\left([u]^p_{p,s_1}\right)(-Δ)^{s_1}_p u + M\left([u]^q_{q,s_2}\right)(-Δ)^{s_2}_q u = λ\big[a(x)|u|^{p-2}u + b(x)|u|^{q-2}u\big] + h(x), & \text{in } Ω, \\ u = 0, & \text{on } \mathbb{R}^N \setminus Ω, \end{cases} \end{equation*}
where $Ω\subset \mathbb{R}^N$ ($N \geq 1$) is a bounded domain with smooth boundary, $0 < s_1 < s_2 < 1$, and $s_1 p < N$. We assume $1 < q \leq p < θp < p^{*}_{s_1} := \dfrac{Np}{N - s_1 p}$, and $λ\in \mathbb{R}$. The functions $a(x), b(x)$, and $h(x)$ are non-negative, with $a, b \in L^\infty(Ω)$ and $h \in L^q(Ω)$.
Using variational methods, we establish the existence of at least two weak solutions. The first solution is obtained via the direct minimization of the associated energy functional, and the second is obtained by applying the Mountain Pass Theorem. We also prove a nonexistence result for small values of the parameter $λ> 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kirchhoff-type equations involving the Fractional $(p,q)-$Laplacian Carrero, Lisbeth Hernández-Llanos, Pedro Analysis of PDEs 35A15, 35J20, 35J60, 35R11 In this paper, we study the existence and nonexistence of solutions for the following Kirchhoff-type fractional $(p\text{-}q)$-Laplacian problem: \begin{equation*} \begin{cases} M\left([u]^p_{p,s_1}\right)(-Δ)^{s_1}_p u + M\left([u]^q_{q,s_2}\right)(-Δ)^{s_2}_q u = λ\big[a(x)|u|^{p-2}u + b(x)|u|^{q-2}u\big] + h(x), & \text{in } Ω, \\ u = 0, & \text{on } \mathbb{R}^N \setminus Ω, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ ($N \geq 1$) is a bounded domain with smooth boundary, $0 < s_1 < s_2 < 1$, and $s_1 p < N$. We assume $1 < q \leq p < θp < p^{*}_{s_1} := \dfrac{Np}{N - s_1 p}$, and $λ\in \mathbb{R}$. The functions $a(x), b(x)$, and $h(x)$ are non-negative, with $a, b \in L^\infty(Ω)$ and $h \in L^q(Ω)$. Using variational methods, we establish the existence of at least two weak solutions. The first solution is obtained via the direct minimization of the associated energy functional, and the second is obtained by applying the Mountain Pass Theorem. We also prove a nonexistence result for small values of the parameter $λ> 0$. |
| title | Kirchhoff-type equations involving the Fractional $(p,q)-$Laplacian |
| topic | Analysis of PDEs 35A15, 35J20, 35J60, 35R11 |
| url | https://arxiv.org/abs/2508.16281 |