Liouville-Type Theorem for Choquard Equation in Half-Space

本文研究半空间上Choquard方程 { −Δu( y )= ∫ ∂ ℝ + N | u( x ¯ ,0 ) | p | ( x ¯ ,0 )−y | N−α d x ¯ | u( y ) | p−2 u( y ),y∈ ℝ + N ∂u ∂ν ( x ¯ ,0 )= ∫ ℝ + N | u( y ) | p | ( x ¯ ,0 )−y | N−α dy | u( x ¯ ,0 ) | p−2 u( x ¯ ,0 ),( x ¯ ,0 )∈∂ ℝ + N 应用积分形式的移动平面法，证明了在参数p的一定取值条件下，该方程不存在非平凡正解。This paper investigates the Choquard equation in a half-space setting { −Δu( y )= ∫ ∂ ℝ + N | u( x ¯ ,0 ) | p | ( x ¯ ,0 )−y | N−α d x ¯ | u( y ) | p−2 u( y ),y∈ ℝ + N ∂u ∂ν ( x ¯ ,0 )= ∫ ℝ + N | u( y ) | p | ( x ¯ ,0 )−y | N−α dy | u( x ¯ ,0 ) | p−2 u( x ¯ ,0 ),( x ¯ ,0 )∈∂ ℝ + N . By applying the method of moving planes in integral form, it is demonstrated that under certain conditions on the parameter p, the equation has no nontrivial positive solutions.