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{${\mathcal{P}}_{\lambda ,\epsilon }$} -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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded open set, $\epsilon \in (0,1]$, $0<s<1<q<p<N$, and $p^{\ast }=\frac{Np}{N-p}$, and $\lambda \in \mathbb{R}$ is a parameter. For $\lambda \le 0$, we show that (${\mathcal{P}}_{\lambda ,\epsilon }$) has no nontrivial solution. For $\lambda >0$, we prove Ambrosetti-Brezis-Cerami type results. In particular, we prove the existence of ${\mathrm{\Lambda }}_{\epsilon }$ such that (${\mathcal{P}}_{\lambda ,\epsilon }$) has a positive minimal solution for $0<\lambda <{\mathrm{\Lambda }}_{\epsilon }$, a positive solution for $\lambda ={\mathrm{\Lambda }}_{\epsilon }$ and no positive solution for $\lambda >{\mathrm{\Lambda }}_{\epsilon }$. We also prove the existence of $0<{\lambda }^{\mathrm{\#}}\le {\mathrm{\Lambda }}_{\epsilon }$ such that (${\mathcal{P}}_{\lambda ,\epsilon }$) has at least two positive solutions for $\lambda \in (0,{\lambda }^{\mathrm{\#}})$ provided $\epsilon$ 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).