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

Abstract

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