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 {eqnarray} {split} -Δ_pu+(-Δ)_p^s u&={λ}{uγ}+u^r in Ω, &>0 in Ω,&=0 in R^n Ω; {split} {eqnarray} where {equation*} (-Δ )_p^s u(x)= cn,sP.V.∫R^n{|u(x)-u(y)|ᵖ⁻²(u(x)-u(y))}{|x-y|ⁿ⁺ˢᵖ} d y, {equation*} and -Δₚ is the usual p-Laplace operator. Under the assumptions that Ω is a bounded domain in Rⁿ with regular enough boundary, $p>1$, $n> p$, s∈(0,1), λ>0 and r∈(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<γ<1 and some certain values of λ. Further, for every γ>0, assuming strict convexity of Ω, for $p=2$ and s∈(0,1/2), we will show the existence of at least two positive weak solutions to the problem, for small values of λ, extending the result of {garaingeometric}. Here cn,s is a suitable normalization constant, and P.V. stands for Cauchy Principal Value.