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

Vinayak Mani Tripathi

Abstract. 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.

Keywords: mixed local and nonlocal operators, Kirchhoff type problem, critical nonlinearity, Nehari manifold.

Mathematics Subject Classification: 35A01, 35A15, 35B33, 35R11.

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