Maximum principles involving the uniformly elliptic nonlocal operator

• Published in: Mathematical methods in the applied sciences 2023-03, Vol.46 (4), p.3721-3740
• Main Authors: Jiayan, Wu, Qu, Meng, Zhang, Jingjing, Zhang, Ting
• Format: Article
• Language: English
• Subjects: Coercivity; Domains; Elliptic functions; Liouville theorem; Maximum principle; maximum principles; method of moving planes; monotonicity; Operators (mathematics); Principles; radial symmetry; Schrodinger equation; uniform elliptic nonlocal operator
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8718

In this paper, we consider equations involving a uniformly elliptic nonlocal operator Aβu(x)=CN,βP.V.∫ℝNa(x−y)(u(x)−u(y))|x−y|N+βdy,$$ {A}_{\beta }u(x)={C}_{N,\beta}\mathrm{P}.\mathrm{V}.\underset{{\mathbb{R}}^N}{\int}\frac{a\left(x-y\right)\left(u(x)-u(y)\right)}{{\left|x-y\right|}^{N+\beta }} dy, $$ where the function a:ℝN↦ℝ$$ a:{\mathbb{R}}^N\mapsto \mathbb{R} $$ is uniformly bounded and radial decreasing. We establish some maximum principles for Aβ$$ {A}_{\beta } $$ in bounded and unbounded domains. Since there is no decay condition in the unbounded domain, we make use of an approximate method to estimate the singular integral to get the maximum principle. As applications of these principles, by carrying out the method of moving planes, we give the monotonicity of solutions to the semilinear equation in the coercive epigraph, which extends the result of Dipierro‐Soave‐Valdinoci [Math. Ann.2017, 369(3‐4): 1283–1326]. Moreover, we obtain the radial symmetry and monotonicity of solutions to the generalized Schrödinger equation in a weaker condition, which is the improvement of the result of Tang [Math. Methods Appl. Sci. 2017, 40(7): 2596–2609]. In addition, the maximum principle also plays an important role in acquiring monotonicity of solutions and a Liouville theorem.