On sign-changing solutions for mixed local and nonlocal $p$-Laplace operator
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| Format: | Preprint |
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2025
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| _version_ | 1866909705846128640 |
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| author | Bhowmick, Souvik Ghosh, Sekhar |
| author_facet | Bhowmick, Souvik Ghosh, Sekhar |
| contents | 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 $p=2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_04361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On sign-changing solutions for mixed local and nonlocal $p$-Laplace operator Bhowmick, Souvik Ghosh, Sekhar Analysis of PDEs 35M12, 35R11, 47J30, 35J60, 35J92 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 $p=2$. |
| title | On sign-changing solutions for mixed local and nonlocal $p$-Laplace operator |
| topic | Analysis of PDEs 35M12, 35R11, 47J30, 35J60, 35J92 |
| url | https://arxiv.org/abs/2503.04361 |