A semilinear equation involving the fractional Laplacian in ℝⁿ

2015/03/07 by Yan Li, Li, Yan
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1503.02224

arxiv created 2015/03/07 · arxiv updated 2015/03/10

Abstract

In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space ℝⁿ: (-Δ)^α/2 u(x) = f(x_n) u^p(x), x ∈ ℝⁿ in the subcritical case with 1<p<(n+α)/(n-α). Instead of carrying out direct investigations on pseudo-differential equation (\refn26), we first seek its equivalent form in an integral equation as below: u(x)=∫_ℝⁿG_∞(x,y) f(y_n) u^p(y) dy, where G_∞(x,y) is the Green's function associated with the fractional Laplacian in ℝⁿ. Exploiting the method of moving planes in integral forms, we are able to derive the nonexistence of positive solutions for (\refn27) in the subcritical case. Hence the same conclusion is true for (\refn26).

Related

• Solutions with various structures for semilinear equations in \mathbb Rⁿ driven by fractional Laplacian
• Existence and symmetry result for Fractional p-Laplacian in\n \ℝⁿ
• Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on ℝⁿ and ℝⁿ₊
• The Pohozaev identity for the fractional Laplacian
• Semilinear equations involving the fractional Laplacian on domains