Global Multiplicity and Comparison Principles for Singular Problems driven by Mixed Local-Nonlocal Operators

TL;DR

The paper addresses a singular mixed local-nonlocal PDE $-\Delta_p u + (-\Delta_q)^s u = \frac{\lambda}{u^{\delta}} + u^r$ in a bounded domain, with $p>sq$ and $0<\delta<1$, establishing a sharp threshold $\Lambda$ that separates regimes of existence, nonexistence, and multiplicity. It develops a global bifurcation-type framework built on a Hopf-type strong comparison principle and a Sobolev-to-Hölder minimizer equivalence, enabling global multiplicity results in subcritical and critical growth settings. Key contributions include the global two-solution multiplicity in subcritical and critical cases, a rigorous SCP adapted to mixed operators with singular nonlinearities, and robust a priori estimates (uniform $L^{\infty}$ and regularity) that underpin the variational analysis. The results extend the theory of mixed local-nonlocal problems, offering tools applicable to a broad class of singular nonlinear PDEs and providing a foundation for further global bifurcation analyses in nonlocal frameworks.

Abstract

We study a singular elliptic problem driven by a mixed local-nonlocal operator of the form \begin{equation*} \begin{aligned} -Δ_p u + (-Δ_q)^s u &= \fracλ{u^δ} + u^r \text{ in } Ω\newline u > 0 \text{ in } Ω,\ u &= 0 \text{ in } \mathbb{R}^N \setminus Ω \end{aligned} \end{equation*} where $p > sq$, $0<δ<1$ and $λ> 0$ is a parameter. The nonlinearity exhibits a singular power-type behavior near zero and displays at most a critical growth at infinity. We establish a global multiplicity result with respect to the parameter $λ$ by identifying a sharp threshold that separates existence, non-existence, and multiplicity regimes, a result that is new for singular problems involving mixed local-nonlocal operators. We also derive a Hopf-type strong comparison principle adapted to this nonlinear setting, which provides the main analytical tool for the global multiplicity result. Additionally, we investigate qualitative properties of solutions that are essential for the variational analysis, such as a uniform $L^{\infty}$-estimate and a Sobolev versus Hölder local minimizer result. The analytical tools developed herein are of independent mathematical interest, with their applicability extending over a broader class of mixed local-nonlocal problems.

Theorems & Definitions (44)

• Definition 1.1
• Theorem 1.2: Existence and non-existence
• Theorem 1.3: Global multiplicity in the subcritical case
• Theorem 1.4: Global multiplicity in the critical case
• Theorem 1.5: Comparison Principle
• Theorem 1.6
• Theorem 2.1: Strong Comparison Principle
• proof
• Proposition 2.2
• Lemma 2.3