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

Abstract

We investigate a perturbed Gelfand problem involving a mixed local-nonlocal $p$-Laplacian operator with singular nonlinearity: \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 $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $λ> 0 $ is a parameter, $0\leq β<1$ 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.