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{$\mathcal{P}_{\lambda,\varepsilon}$} -\Delta_p u+\varepsilon(-\Delta_p)^s u=\lambda|u|^{q-2}u+|u|^{p^*-2}u \;\text{ in }\Omega,\quad u=0 \; \text{ in }\mathbb{R}^N \setminus \Omega, \end{equation} where $\Omega\subset\mathbb{R}^N$ is a bounded open set, $\epsilon\in(0,1]$, $0<s<1<q<p<N$, and $p^*=\frac{Np}{N-p}$, and $\lambda \in \mathbb{R}$ is a parameter. For $\lambda \leq 0$, we show that ($\mathcal{P}_{\lambda,\varepsilon}$) has no nontrivial solution. For $\lambda>0$, we prove Ambrosetti-Brezis-Cerami type results. In particular, we prove the existence of $\Lambda_\varepsilon$ such that ($\mathcal{P}_{\lambda,\varepsilon}$) has a positive minimal solution for $0<\lambda<\Lambda_\varepsilon$, a positive solution for $\lambda=\Lambda_\varepsilon$ and no positive solution for $\lambda>\Lambda_\varepsilon$. We also prove the existence of $0<\lambda^\#\leq\Lambda_\varepsilon$ such that ($\mathcal{P}_{\lambda,\varepsilon}$) has at least two positive solutions for $\lambda\in(0,\lambda^\#)$ 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).