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

TL;DR

This paper studies a mixed local and nonlocal Laplacian equation of Lane-Emden type, revealing the absence of a traditional critical exponent and establishing solution existence near the Sobolev critical value.

Contribution

It introduces a novel elliptic equation combining local and fractional Laplacians, demonstrating the failure of the duality of critical exponents.

Findings

Existence of solutions below the Sobolev critical exponent.

No traditional critical exponent due to operator interaction.

First example showing duality failure of critical exponents.

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.