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

Abstract

<jats:p>In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem \[\begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\text{ in }\Omega, \\ u=0 &\text{ on }\mathbb R^N\setminus \Omega, \end{cases}\] 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.</jats:p>

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