Integrability, regularity and symmetry of positive integrable solutions for Wolff type integral systems.

In this paper, we are concerned with the optimal integrability, regularity and symmetry of integrable solutions for the following Wolff type integral systems: (1.1) { u (x) = R 1 (x) W β , γ (v q) (x) , u (x) > 0 , x ∈ R N , v (x) = R 2 (x) W β , γ (u p) (x) , v (x) > 0 , x ∈ R N , where γ > 2 , β > 0 , β γ < N , p , q > γ − 1 with γ − 1 p + γ − 1 + γ − 1 q + γ − 1 = N − β γ N , R 1 , R 2 are double bounded in R N and W β , γ (h) (x) : = ∫ 0 ∞ [ ∫ B t (x) h (y) d y t N − β γ ] 1 γ − 1 d t t. Firstly, we prove the optimal integrability and boundedness of solutions (u , v) ∈ L p + γ − 1 (R N) × L q + γ − 1 (R N) for system (1.1) , by constructing a nonlinear, contracting operator and applying the regularity lifting lemma of C. Ma, W. Chen and C. Li (2011) [21]. Moreover, we exploit the general regularity lifting theorem to derive the Lipschitz continuity of u and v when R 1 ≡ R 2 ≡ 1 in R N. These extend the above important results of C. Ma, W. Chen and C. Li to γ > 2. We also prove that u and v vanish at infinity. As the corollaries of the above results, we obtain the optimal integrability, boundedness and the property of vanishing at infinity of integrable solutions for the corresponding γ -Laplace and k -Hessian systems. Secondly, we use the method of moving planes in integral forms to prove the symmetry and monotonicity of solutions (u , v) ∈ L p + γ − 1 (R N) × L q + γ − 1 (R N) for system (1.1) when R 1 ≡ R 2 ≡ 1 in R N , which extends the useful result of W. Chen and C. Li (2011) [5] to γ > 2. In comparison with the above two papers, Minkowski's inequality is crucial in our proofs. We believe that our arguments can be used to prove similar results for other Wolff type integral systems when γ > 2. [ABSTRACT FROM AUTHOR]