Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions
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2025
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| author | Bhakta, Mousomi Biswas, Nirjan Das, Paramananda |
| author_facet | Bhakta, Mousomi Biswas, Nirjan Das, Paramananda |
| contents | We study the existence and multiplicity of positive solutions for the following concave-critical problem driven by an operator of mixed order obtained by the sum of the classical $p$-Laplacian and of the fractional $p$-Laplacian, \begin{equation}\tag{$\mathcal{P}_{λ,\varepsilon}$}
-Δ_p u+\varepsilon(-Δ_p)^s u=λ|u|^{q-2}u+|u|^{p^*-2}u \;\text{ in }Ω,\quad
u=0 \; \text{ in }\mathbb{R}^N \setminus Ω, \end{equation} where $Ω\subset\mathbb{R}^N$ is a bounded open set, $ε\in(0,1]$, $0<s<1<q<p<N$, and $p^*=\frac{Np}{N-p}$, and $λ\in \mathbb{R}$ is a parameter. For $λ\leq 0$, we show that ($\mathcal{P}_{λ,\varepsilon}$) has no nontrivial solution. For $λ>0$, we prove Ambrosetti-Brezis-Cerami type results. In particular, we prove the existence of $Λ_\varepsilon$ such that ($\mathcal{P}_{λ,\varepsilon}$) has a positive minimal solution for $0<λ<Λ_\varepsilon$, a positive solution for $λ=Λ_\varepsilon$ and no positive solution for $λ>Λ_\varepsilon$. We also prove the existence of $0<λ^\#\leqΛ_\varepsilon$ such that ($\mathcal{P}_{λ,\varepsilon}$) has at least two positive solutions for $λ\in(0,λ^\#)$ provided $\varepsilon$ small enough. This extends the recent result of Biagi and Vecchi (Nonlinear Anal. 256 (2025),113795), Amundsen, et al. (Commun. Pure Appl. Anal., 22(10):3139-3164, 2023) from $p=2$ to the general $1<p<N$. Additionally, it extends the classical result of Azorero and Peral (Indiana Univ. Math. J., 43(3):947-957, 1994) to the mixed local-nonlocal quasilinear problems. Moreover, our results complements the multiplicity results for nonnegative solutions in da Silva, et al. (J. Differential Equations, 408:494-536, 2024). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15000 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions Bhakta, Mousomi Biswas, Nirjan Das, Paramananda Analysis of PDEs 35B09, 35B33, 35J20, 35J25, 35J62, 35M12 We study the existence and multiplicity of positive solutions for the following concave-critical problem driven by an operator of mixed order obtained by the sum of the classical $p$-Laplacian and of the fractional $p$-Laplacian, \begin{equation}\tag{$\mathcal{P}_{λ,\varepsilon}$} -Δ_p u+\varepsilon(-Δ_p)^s u=λ|u|^{q-2}u+|u|^{p^*-2}u \;\text{ in }Ω,\quad u=0 \; \text{ in }\mathbb{R}^N \setminus Ω, \end{equation} where $Ω\subset\mathbb{R}^N$ is a bounded open set, $ε\in(0,1]$, $0<s<1<q<p<N$, and $p^*=\frac{Np}{N-p}$, and $λ\in \mathbb{R}$ is a parameter. For $λ\leq 0$, we show that ($\mathcal{P}_{λ,\varepsilon}$) has no nontrivial solution. For $λ>0$, we prove Ambrosetti-Brezis-Cerami type results. In particular, we prove the existence of $Λ_\varepsilon$ such that ($\mathcal{P}_{λ,\varepsilon}$) has a positive minimal solution for $0<λ<Λ_\varepsilon$, a positive solution for $λ=Λ_\varepsilon$ and no positive solution for $λ>Λ_\varepsilon$. We also prove the existence of $0<λ^\#\leqΛ_\varepsilon$ such that ($\mathcal{P}_{λ,\varepsilon}$) has at least two positive solutions for $λ\in(0,λ^\#)$ provided $\varepsilon$ small enough. This extends the recent result of Biagi and Vecchi (Nonlinear Anal. 256 (2025),113795), Amundsen, et al. (Commun. Pure Appl. Anal., 22(10):3139-3164, 2023) from $p=2$ to the general $1<p<N$. Additionally, it extends the classical result of Azorero and Peral (Indiana Univ. Math. J., 43(3):947-957, 1994) to the mixed local-nonlocal quasilinear problems. Moreover, our results complements the multiplicity results for nonnegative solutions in da Silva, et al. (J. Differential Equations, 408:494-536, 2024). |
| title | Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions |
| topic | Analysis of PDEs 35B09, 35B33, 35J20, 35J25, 35J62, 35M12 |
| url | https://arxiv.org/abs/2504.15000 |