Mixed Local-Nonlocal Operators and Singularity: A Multiple-Solution Perspective

TL;DR

The paper addresses the multiplicity of positive solutions for a Dirichlet problem driven by a mixed local-nonlocal operator $-\Delta_p u + (-\Delta_p)^s u$ with a singular nonlinearity $\lambda \frac{f(u)}{u^{\beta}}$ on a bounded domain. It develops a sub- and supersolution framework to prove the existence of a positive solution for all $\lambda>0$ and at least two positive solutions for $\lambda$ in a certain interval $(\lambda_*, \lambda^*)$, with a third solution obtained in two special cases: the non-singular case ($\beta=0$) and the linear singular case ($p=2$, $0<\beta<1$) using Amann's fixed point theorem together with a Hopf-type strong comparison principle. A novel aspect is the construction of sub-/supersolutions leveraging the homogeneity of the mixed operator, and the establishment of a Hopf-type SCP for singular linear mixed problems, which underpins the third-solution results. These contributions advance the theory of multiplicity for mixed local-nonlocal operators and broaden the framework for nonlinear PDEs with singular nonlinearities while highlighting open questions for the general non-linear singular regime ($p\neq 2$, $0<\beta<1$).

Abstract

We investigate the existence of multiple positive solutions for the following Dirichlet boundary value problem: \begin{equation*} \begin{aligned} -Δ_p u + (-Δ_p)^s u = λ\frac{f(u)}{u^β}\ \text{in} \ Ω\newline u >0\ \text{in} \ Ω,\ u =0\ \text{in} \ \mathbb{R}^N \setminus Ω\end{aligned} \end{equation*} where $Ω$ is an arbitrary bounded domain in $\mathbb{R}^N$ with smooth boundary, $0\leq β<1$ and $f$ is a non-decreasing $C^1$-function which is $p$-sublinear at infinity and satisfies $f(0)>0$. By employing the method of sub- and supersolutions, we establish the existence of a positive solution for every $λ>0$ and that of two positive solutions for a certain range of the parameter $λ$. In the non-singular case (i.e. when $β=0$) and in the linear case with singularity (i.e. when $p=2$ and $0<β<1$), we further apply Amann's fixed point theorem to show that the problem admits at least three positive solutions within this range of $λ$. The mixed local-nonlocal nature of the operator and the non-linearity pose challenges in constructing sub- and supersolutions, however, these are effectively addressed through the operator's homogeneity.

Theorems & Definitions (26)

• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof