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} -\Delta_pu+(-\Delta)_p^s u&=\frac{\lambda}{u^{\gamma}}+u^r \text { in } \Omega, \\u&>0 \text{ in } \Omega,\\u&=0 \text { in }\mathbb{R}^n \backslash \Omega; \end{split} \end{eqnarray} where \begin{equation*} (-\Delta )_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 $-\Delta_p$ is the usual $p$-Laplace operator. Under the assumptions that $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with regular enough boundary, $p>1$, $n> p$, $s\in(0,1)$, $\lambda>0$ and $r\in(p-1,p^*-1)$ where $p^*$ 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 $\Omega$, for $p=2$ and $s\in(0,1/2)$, 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 $\operatorname{P.V.}$ stands for Cauchy Principal Value.