Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth

In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem \[{cases}M(\ ∫R^N|∇ u|^2dx+_{R2N}{|u(x)-u(y)|^2}{|x-y|N+2s}dxdy)L(u) \\ = λ {f(x)}|u|ᵖ⁻²u+|u|2^*-2u & in Ω, \\ u=0 & on R^N Ω, {cases}\] where \(Ω^N\) is bounded domain with smooth boundary, \(1< p< 2< 2^*=2N/N-2\), \(N≥ 3\), \(λ> 0\), \(M\) is a Kirchhoff coefficient and \(L\) denotes the mixed local and nonlocal operator. The weight function \(f∈ L2^*/2^*-p(Ω)\) is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions.