Abstract:

This article focuses on a class of local and nonlocal elliptic equations with Nirenberg-Brezis problem

$\begin{equation*} \left\{\begin{array}{ll} - \Delta u +(-\Delta)^su= \lambda u+ |u|^{2^*-2}u,~~ & x\in \Omega,\\ u=0, & x\in \mathbb{R}^N\setminus \Omega, \end{array}\right. \end{equation*}$

where $ \Omega $ is a bounded smooth domain of $ \mathbb{R}^N $ $ (N>2) $, $ s\in (0,1) $, $ 2^*= \frac{2N}{N-2} $. The above problem has at least one positive solution for $ \lambda\in (\lambda^*,\lambda_1) $ with $ \lambda^* \in\left[\lambda_{1, s}, \lambda_1\right) $, and has no positive solutions for $ \lambda\in [\lambda_1,+\infty) $, where $ \lambda_{1,s} $ and $ \lambda_1 $ is the first eigenvalue of Dirichlet boundary problem of operator $ (-\Delta)^s $ and $ - \Delta +(-\Delta)^s $, respectively. Firstly, we estimate the lower boundedness of $ \lambda^* $. Then, by establishing proper linking sets and applying Willem' linking principle, we prove the existence of nodal solution for $ \lambda\in [\lambda_1,+\infty) $.

Key words: local and nonlocal operator, critical exponents, linking