Abstract
In this paper we consider the following Hardy-Littlewood-Sobolev (HLS)-type system of nonlinear equations in the half-space R + n : u(x)= ∫ R + n ( 1 | x − y | n − α − 1 | x ∗ − y | n − α ) v q (y)dy, v(x)= ∫ R + n ( 1 | x − y | n − α − 1 | x ∗ − y | n − α ) u p (y)dy, where p,q>1 and x ∗ is the reflection of x about the boundary { x n =0}. By using the method of moving planes in integral forms, we obtain monotonicity of the positive solution of the integral equations system of the abstract in three cases: the so-called subcritical, critical, and supercritical cases, and we obtain a new Liouville-type theorem of this system under some integrability conditions. In particular, our results unify and generalize many cases of Liouville-type theorems in (Cao and Dai in J. Math. Anal. Appl. 389:1365-1373, 2012; Cao and Dai in J. Inequal. Appl. 2013:37, 2013) and (Li et al. in Complex Var. Elliptic Equ. 2013, doi:10.1080/17476933.2013.854346).MSC:35B05, 35B45.
Highlights
In [ ], Chen and Li discussed the HLS-type system of nonlinear equations in the whole space Rn: u(x) = Rn |x–y|n–α vq|x–y|n–α up dy, x ∈ Rn. ( . )By the method of moving planes in integral forms they derived that the positive solutions of ( . ) are radially symmetric and such solutions are nonexistent under some integrability conditions.In a recent paper of Chen and Li [ ], the equivalence between integral equation ( . ) and the following PDEs was established: (– ) α u(x) =
1 Introduction In [ ], Chen and Li discussed the HLS-type system of nonlinear equations in the whole space Rn: u(x) =
We prove that the positive solution pair (u, v) of ( . ) is strictly monotonically increasing with respect to the variable xn
Summary
|x–y|n–α up dy, x ∈ Rn. By the method of moving planes in integral forms they derived that the positive solutions of In this paper we want to generalize monotonicity and nonexistence results of positive solutions of an HLS-type system in the whole space Rn to ones in a half-space. Suppose that u ∈ Lp (Rn+) and v ∈ Lq (Rn+) is a pair of positive solutions of integral system We will use the method of moving planes in integral forms to obtain the monotonicity of the positive solutions of system Concerning monotonicity and nonexistence of solutions is true in all three cases: subcritical, critical, and supercritical. Unifies and generalizes some Liouville-type results of positive solutions of other integral systems. We need the equivalent form of the Hardy-Littlewood-Sobolev inequality
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