To this end, we employ a certain type of Kelvin transform and the method of moving planes in integral forms.
In order to prove the Liouville theorem, Theorem 1.2, we carry out the method of moving planes in integral form in the positive (x_{n}) direction.
In this section, by the method of moving planes in integral forms we derive the nonexistence of positive solutions to the integral system (1.4) and obtain a new Liouville-type theorem in a half-space.
In this section, by using the method of moving planes in integral forms, we derive the nonexistence of positive solutions to integral system (1.2) and obtain a new Liouville-type theorem on a half-space.
By the method of moving planes in integral forms they derived that the positive solutions of (1.1) are radially symmetric and such solutions are nonexistent under some integrability conditions.
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.
Assume that u ∈ L p + 1 ( R + n ) and v ∈ L q + 1 ( R + n ) are nonnegative, then u = v ≡ 0. In this paper, we further consider the nonnegative solution of the integral equations system (1.4) by using the method of moving planes in integral forms.
Assume that u ∈ L p 1 ( R + n ) and v ∈ L q 1 ( R + n ) are nonnegative, then u = v ≡ 0. To prove Theorem 1.4, we will use the method of moving planes in integral forms to obtain the monotonicity of the positive solutions of system (1.4).
By the method of moving plane in integral form, Jin and Li in [5] showed that all of the positive solutions of (1.7) are radially symmetric.
To obtain our results, in this section, we will use the method of moving plane in the integral forms recently introduced by Chen et al. in [13] and prove the radial symmetry and monotonicity of positive solutions of system (1.1).
