Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915393489076224 |
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| author | Barrios, Begoña Del Pezzo, Leandro M. Quaas, Alexander |
| author_facet | Barrios, Begoña Del Pezzo, Leandro M. Quaas, Alexander |
| contents | In this paper, we investigate a mixed elliptic equation involving both local and nonlocal Laplacian operators, with a power-type nonlinearity. Specifically, we consider a Lane-Emden type equation of the form \[-Δu + (-Δ)^s u = u^p,\quad\mbox{ in }\mathbb{R}^n.\] where the operator combines the classical Laplacian and the fractional Laplacian. We establish the existence of solutions for exponents slightly below the critical local Sobolev exponent, that is, for $p < \frac{n+2}{n-2}$, with $p$ close to $\frac{n+2}{n-2}$.
Our results show that, due to the interaction between the local and nonlocal operators, this mixed Lane-Emden-Fowler equation does not admit a critical exponent in the traditional sense. The existence proof is carried out using a Lyapunov-Schmidt type reduction method and, as far as we know, provide the first example of an elliptic operator for which the duality between critical exponents fails. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12258 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation Barrios, Begoña Del Pezzo, Leandro M. Quaas, Alexander Analysis of PDEs In this paper, we investigate a mixed elliptic equation involving both local and nonlocal Laplacian operators, with a power-type nonlinearity. Specifically, we consider a Lane-Emden type equation of the form \[-Δu + (-Δ)^s u = u^p,\quad\mbox{ in }\mathbb{R}^n.\] where the operator combines the classical Laplacian and the fractional Laplacian. We establish the existence of solutions for exponents slightly below the critical local Sobolev exponent, that is, for $p < \frac{n+2}{n-2}$, with $p$ close to $\frac{n+2}{n-2}$. Our results show that, due to the interaction between the local and nonlocal operators, this mixed Lane-Emden-Fowler equation does not admit a critical exponent in the traditional sense. The existence proof is carried out using a Lyapunov-Schmidt type reduction method and, as far as we know, provide the first example of an elliptic operator for which the duality between critical exponents fails. |
| title | Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.12258 |