Multiplicity of solutions for mixed local-nonlocal elliptic equations with singular nonlinearity

We will prove multiplicity results for the mixed local-nonlocal elliptic equation of the form \begin{eqnarray} \begin{split} -Δ_pu+(-Δ)p^s u&=\fracλ{u^γ}+u^r \text { in } Ω, \u&>0 $\text{ in }$ Ω,\u&=0 \text { in }$\mathbb{R}$^n \backslash Ω; \end{split} \end{eqnarray} where \begin{equation*} (-Δ)p^s u(x)= c{n,s}\operatorname{P.V.}\int{$\mathbb{R}$^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}} d y, \end{equation*} and $-{\mathrm{\Delta }}_p$ is the usual $p$-Laplace operator. Under the assumptions that $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^n$ with regular enough boundary, $p>1$, $n>p$, $s\in (0,1)$, $\lambda >0$ and $r\in (p-1,p^{\ast }-1)$ where $p^{\ast }$ is the critical Sobolev exponent, we will show there exist at least two weak solutions to our problem for $0<\gamma <1$ and some certain values of $\lambda$. Further, for every $\gamma >0$, assuming strict convexity of $\mathrm{\Omega }$, for $p=2$ and $s\in (0,1\mathrm{/}2)$, we will show the existence of at least two positive weak solutions to the problem, for small values of $\lambda$, extending the result of \cite{garaingeometric}. Here $c_{n,s}$ is a suitable normalization constant, and $\mathrm{P.V.}$ stands for Cauchy Principal Value.