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arXiv · 2507.12258

Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation

Begoña Barrios·Leandro M. Del Pezzo·Alexander Quaas
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Abstract

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 \[-\Delta u + (-\Delta)^s u = u^p,\quad\mbox{ in }\mathbb{R}^n.\] 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.

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Begoña Barrios, Leandro M. Del Pezzo, Alexander Quaas. 2025-07-16. Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation. https://arxiv.org/abs/2507.12258

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