We prove the multiplicity of solutions for the mixed local-nonlocal elliptic equation of the form \(\begin{aligned} \begin{aligned} -\varDelta _pu+(-\varDelta )_p^s u&=\frac{\lambda }{u^{\gamma }}+u^r \text{ in } \varOmega , \\ u&>0 \text { in } \varOmega ,\\ u&=0 \text{ in } \mathbb {R}^n \backslash \varOmega ; \end{aligned} \end{aligned}\) where \(\begin{aligned} (-\varDelta )_p^s u(x):= c_{n,s}\operatorname {P.V.}\displaystyle \int _{\mathbb {R}^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}} \,d y. \end{aligned}\) Under the assumptions that \(\varOmega \) is a \(C^1\) bounded domain in \(\mathbb {R}^{n}\) , \(1<p<n\) , \(s\in (0,1)\) , \(\lambda >0\) and \(r\in (p-1,p^*-1)\) where \(p^*\) is the critical Sobolev exponent, we exhibit at least two weak solutions to our problem for \(0<\gamma <1\) and for small \(\lambda \) . The main novelty of this work is that, in the linear case \(p=2\) , we extend the range \(\gamma >1\) under the additional assumptions that \(\varOmega \) is a strictly convex domain, with smooth boundary \(\partial \varOmega \) and \(s\in (0,1/2)\) .
