Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions

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).