A Hopf type lemma for nonlocal pseudo-relativistic equations and its applications

Abstract

In this paper, we consider the nonlinear equation involving the nonlocal pseudo-relativistic operators $(-\mathrm{\Delta }+m^2{)}^{s}u(x)=f(x,u(x)),$ where 0<s<1 and mass m>0. The nonlocal pseudo-relativistic operator includes the pseudo-relativistic Schrödinger operator $\sqrt{-\mathrm{\Delta }+m^2}$. When $m\to 0^{+}$, the nonlocal pseudo-relativistic operator $(-\mathrm{\Delta }+m^2{)}^s$ is also closely related to the fractional Laplacian operator $(-\mathrm{\Delta }{)}^s$. But these two operators are quite different. We first establish a Hopf type lemma for anti-symmetric functions to nonlocal pseudo-relativistic operators, which play a key role in the method of moving planes. The main difficulty is to construct a suitable sub-solution to nonlocal pseudo-relativistic operators. Then we prove a pointwise estimate to nonlocal pseudo-relativistic operators. As an application, combined with the Hopf type lemma and the pointwise estimate, we obtain the radial symmetry and monotonicity of positive solutions to the above nonlinear nonlocal pseudo-relativistic equation in the whole space. We believe that the Hopf type lemma will become a powerful tool in applying the method of moving planes on nonlocal pseudo-relativistic equations to obtain qualitative properties of solutions.

Keywords:

• Nonlocal pseudo-relativistic equation
• Hopf type lemma
• method of moving planes
• radial symmetry
• monotonicity

AMS Subject Classifications:

• 35A09
• 35B09
• 35R11
• 35J60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author was supported by the National Natural Science Foundation of China [grant number 12101530], Scientific and Technological Key Projects of Henan Province [grant number 232102310321] and Nanhu Scholars Program for Young Scholars of XYNU [grant number 2023]