Existence and Convergence of Least-Energy Solutions Involving the Logarithmic Schrödinger Operator
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910014825824256 |
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| author | Chen, Huyuan Chen, Rui Hua, Bobo |
| author_facet | Chen, Huyuan Chen, Rui Hua, Bobo |
| contents | In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schrödinger operator and its fractional pseudo-relativistic counterpart, both arising in quantum models with nonlocal and relativistic effects. We first establish the existence, uniqueness, and regularity of weak solutions to equations involving the logarithmic operator \((I - Δ)^{\ln}\) with subcritical logarithmic nonlinearities. We then investigate a Brezis--Nirenberg-type problem involving the fractional pseudo-relativistic Schrödinger operator \((I - Δ)^s\), and prove the existence of least-energy solutions under both subcritical and critical nonlinearities. In particular, we show that these least-energy solutions converge, up to a subsequence, to a nontrivial least-energy solution of the limiting problem as \(s \to 0^+\). Our approach relies on variational methods, including the geometry of the Nehari manifold, uniform positive lower bounds, mountain-pass structure, the Palais--Smale condition, and delicate asymptotic analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07240 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and Convergence of Least-Energy Solutions Involving the Logarithmic Schrödinger Operator Chen, Huyuan Chen, Rui Hua, Bobo Analysis of PDEs 35J60, 35R11, 35A15, 35B33, 47G20 In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schrödinger operator and its fractional pseudo-relativistic counterpart, both arising in quantum models with nonlocal and relativistic effects. We first establish the existence, uniqueness, and regularity of weak solutions to equations involving the logarithmic operator \((I - Δ)^{\ln}\) with subcritical logarithmic nonlinearities. We then investigate a Brezis--Nirenberg-type problem involving the fractional pseudo-relativistic Schrödinger operator \((I - Δ)^s\), and prove the existence of least-energy solutions under both subcritical and critical nonlinearities. In particular, we show that these least-energy solutions converge, up to a subsequence, to a nontrivial least-energy solution of the limiting problem as \(s \to 0^+\). Our approach relies on variational methods, including the geometry of the Nehari manifold, uniform positive lower bounds, mountain-pass structure, the Palais--Smale condition, and delicate asymptotic analysis. |
| title | Existence and Convergence of Least-Energy Solutions Involving the Logarithmic Schrödinger Operator |
| topic | Analysis of PDEs 35J60, 35R11, 35A15, 35B33, 47G20 |
| url | https://arxiv.org/abs/2502.07240 |