Multiple positive solutions to a perturbed Gelfand problem involving mixed local-nonlocal operators and singular nonlinearity

TL;DR

This paper proves the existence of multiple solutions for a complex perturbed Gelfand problem involving mixed local and nonlocal operators with singular nonlinearities, using innovative sub-supersolution methods and a new comparison principle.

Contribution

It introduces a novel sub- and supersolution approach that avoids ODE and Green's function techniques, and establishes the first Hopf-type comparison principle for such operators.

Findings

Existence of multiple solutions for the problem.

Development of a new sub-supersolution construction method.

First Hopf-type comparison principle for mixed local-nonlocal operators.

Abstract

We investigate a perturbed Gelfand problem involving a mixed local-nonlocal $p$-Laplacian operator with singular nonlinearity: \begin{equation*} \begin{aligned} -\Delta_p u + (-\Delta_p)^s u = \lambda \frac{f(u)}{u^{\beta}}\ \text{in} \ \Omega\newline u >0\ \text{in} \ \Omega,\ u =0\ \text{in} \ \mathbb{R}^N \setminus \Omega \end{aligned} \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $\lambda > 0 $ is a parameter, $0\leq \beta <1$ and $f$ is a non-decreasing $C^1$-function with $f(0)>0$. Using the method of sub- and supersolutions, we present a novel multiplicity result and, in specific cases, we also prove a three-solution theorem using Amann's fixed point theorem. Our construction of sub-supersolutions avoids the conventional reliance on ODE techniques and Green's function estimates, thereby making it more adaptable to the nonlinear and nonlocal framework. Additionally, we establish a Hopf-type Strong Comparison Principle for the linear operator with singular nonlinearity, marking the first result of its kind for mixed local-nonlocal operators. This result is crucial in deriving a third solution and holds broader mathematical significance.