Title: On sign-changing solutions for mixed local and nonlocal $p$-Laplace operator Show Chinese title

Title: 关于混合局部和非局部$p$-拉普拉斯算子的变号解

Abstract: In this paper, we use the method of invariant sets of descending flows to demonstrate the existence of multiple sign-changing solutions for a class of elliptic problems with zero Dirichlet boundary conditions. By combining Nehari manifold techniques with a constrained variational approach and Brouwer degree theory, we establish the existence of a least-energy sign-changing solution. Furthermore, we prove that the energy of the least energy sign-changing solution is strictly greater than twice the ground state energy. This work extends the celebrated results of Bartsch $et~al.$ [Proc. Lond. Math. Soc. (3), 91(1): 129-152, 2005] and Chang $et~al.$ [Adv. Nonlinear Stud., 19(1): 29-53, 2019] to the mixed local and nonlocal $p$-Laplace operator, providing a novel contribution even in the case when $p=2$. Show Chinese abstract

Abstract: 在本文中，我们使用下降流不变集的方法，证明了一类具有零狄利克雷边界条件的椭圆问题存在多个变号解。 通过结合Nehari流形技术与约束变分方法和布劳威尔度理论，我们建立了最小能量变号解的存在性。 此外，我们证明了最小能量变号解的能量严格大于基态能量的两倍。 这项工作将Bartsch $et~al.$ [Proc. Lond. Math. Soc. (3), 91(1): 129-152, 2005] 和Chang $et~al.$ [Adv. Nonlinear Stud., 19(1): 29-53, 2019] 的著名结果扩展到混合局部和非局部 $p$-拉普拉斯算子，即使在 $p=2$的情况下也提供了新的贡献。