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

Abstract

In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem $\textcolor{white}\$ \begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\text{ in }\Omega, \\ u=0 &\text{ on }\mathbb R^N\setminus \Omega, \qquad \qquad \end{cases} \textcolor{white}\$$ where $\Omega\subset\mathbb{R}^N$ is bounded domain with smooth boundary, $1\lt p\lt 2\lt 2^*=\frac{2N}{N-2}$, $N\geq 3$, $\lambda\gt 0$, $M$ is a Kirchhoff coefficient and $\mathcal{L}$ denotes the mixed local and nonlocal operator. The weight function $f\in L^{\frac{2^*}{2^*-p}}(\Omega)$ is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions.