Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators

Abstract

In this paper, we consider the uniformly elliptic nonlocal operators Aαu(x) = Cn,αP.V. R n a(x − y)(u(x) − u(y)) |x − y| n+α dy,

Maximum principles involving the uniformly elliptic nonlocal operator

“…Remark It is worth noting that the De Giorgi‐type nonlinearity $$$ f(u)=u-{u}^3 $$$ satisfies condition (). Our assumption () is more general than those assumed in Berestycki‐Hamel‐Monneau, 36 Chen‐Bao‐Li, 37 Dipierro‐Soave‐Valdinoci, 29 Liu, 38 Qu‐Wu‐Zhang, 26 Wu‐Chen 39,40 for Laplacian $$$ -\Delta $$$ and fractional Laplacian $$$ {\left(-\Delta \right)}^s $$$, in which $$$ f\left(\cdotp \right) $$$ is required to be nonincreasing on $$$ \left[-1,-1+\delta \right] $$$ and $$$ \left[1-\delta, 1\right] $$$.…”

“…Proof of Theorem 3.5. Let T be any hyperplane, Σ be the half space on one side of the plane T. Set w(x) = u(x) − u(x) for all x ∈ Σ, where x is the reflection of x with respect to T. We define Σ + = {x ∈ Σ|w Our assumption (3.40) is more general than those assumed in Berestycki-Hamel-Monneau, 36 Chen-Bao-Li, 37 Dipierro-Soave-Valdinoci, 29 Liu, 38 Qu-Wu-Zhang, 26 Wu-Chen and hence, we can derive from Theorem 1.1 immediately that w 𝜆 * (x) = 0 almost everywhere in R N , which contradicts assumption (3.39). Thus, w 𝜆 (x) ∶= u(x 𝜆 ) − u(x) < 0 in Σ 𝜆 for all 𝜆 with |𝜆| > M.…”

“…Wang‐Wang 25 obtained monotonicity, symmetry, and nonexistence of the system involving the uniformly elliptic operator $$$ {A}_{\beta } $$$. Recently, based on the sliding method, Qu‐Wu‐Zhang 26 investigated the monotonicity of semilinear equations involving the uniformly elliptic nonlocal operator $$$ {A}_{\beta } $$$, where $$$ a(x) $$$ is positively uniform bounded satisfying a cylindrical condition. Recently, Dai‐Qin‐Wu 21 established various maximum principles for antisymmetric functions involving the pseudo‐relativistic Schrödinger operators $$$ {\left(-\Delta &#x0002B;{m}&#x0005E;2\right)}&#x0005E;s $$$.…”

Qualitative properties of solutions for dual parabolic equation involving uniformly elliptic nonlocal operator

“…Wang and Wang [12] have presented the nonexistence of the system involving the uniform elliptic operator. Recently, Qu et al [13] introduced the monotonicity of semilinear equations involving the uniform elliptic operator $$$ {A}_s $$$ by the sliding method. Recently, Jiayan et al [14] presented the monotonicity of generalized Schrodinger equation involving the operator $$$ {A}_s $$$ in a weaker condition.…”

Sliding method for equations with uniformly parabolic fractional operators

“…In this paper, we will adopt a sliding method to handle the uniformly parabolic fractional operators. As to the uniformly elliptic operators, we can refer to [19]. The sliding method was developed by Berestycki and Nirenberg [1].…”