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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2507.12258 |
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- 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.