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 (\ ₑ₍| u|²dx+ₑ^₂₍|u (x) -u (y) |²|x-y|^{N+2s}dxdy) L (u) \\ = f (x) |u|^p-2u+|u|^2^*-2u & in, \\ u=0 & on RN, cases\ where \ (N\) is bounded domain with smooth boundary, \ (1 p 2 2^*=2NN-2\), \ (N 3\), \ (0\), \ (M\) is a Kirchhoff coefficient and \ (L\) denotes the mixed local and nonlocal operator. The weight function \ (f L^2^*{2^*-p} () \) is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions.