Year: 2026
Type: article
Abstract: In this paper, we develop a method of invariant sets of descending flows to demonstrate the existence of a sign-changing solution for a class of elliptic problems involving mixed local and nonlocal operators with zero Dirichlet boundary conditions. In particular, we prove the existence of infinitely many sign-changing solutions. Next, we employ the Nehari manifold techniques with a constrained variational approach and Brouwer degree theory to establish the existence of a least-energy sign-changing solution and a ground state solution. Moreover, we prove that the energy for the least-energy sign-changing solution is strictly greater than twice the ground-state energy. The results obtained in this paper are completely new in the context of mixed local and nonlocal operators, even for the linear case $ p = 2 $. Specifically, 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.
Authors: Souvik Bhowmick, Sekhar Ghosh
Institution: National Institute of Technology Calicut
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7168359860