On sign-changing solutions for mixed local and nonlocal $ p $ -Laplace operator
- 1National Institute of Technology Calicut , Kozhikode , 673601 , India
- 2National Institute of Technology Calicut, Kozhikode, 673601, India
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.
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BibTeX
@article{bhowmick2026signchanging,
title = {{On sign-changing solutions for mixed local and nonlocal $ p $ -Laplace operator}},
author = {Souvik Bhowmick and Sekhar Ghosh},
journal = {Discrete and Continuous Dynamical Systems - S},
year = {2026},
month = jul,
doi = {10.3934/dcdss.2026184},
url = {https://doi.org/10.3934/dcdss.2026184},
note = {CC BY},
}
RIS
TY - JOUR
TI - On sign-changing solutions for mixed local and nonlocal $ p $ -Laplace operator
AU - Souvik Bhowmick
AU - Sekhar Ghosh
PY - 2026
DA - 2026/07/15
JO - Discrete and Continuous Dynamical Systems - S
DO - 10.3934/dcdss.2026184
UR - https://doi.org/10.3934/dcdss.2026184
AB - 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.
SN - 1937-1179
SN - 1937-1632
ER -