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Existence and nonexistence of positive solutions to an integral system involving Wolff potential

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  • In this paper, we are concerned with the sufficient and necessary conditions for the existence and nonexistence of the positive solutions of the following system involving Wolff type potential: \begin{eqnarray} & u(x) =c_{1}(x)W_{\beta,\gamma}(v^{q})(x), \\ &v(x) =c_{2}(x)W_{\alpha,\tau}(u^{p})(x). \end{eqnarray} Here $x\in R^{n}$, $1 < \gamma,\tau \leq 2$, $\alpha,\beta > 0$, $0< \beta\gamma$, $\alpha \tau < n $, and the functions $c_{1}(x),c_{2}(x)$ are double bounded. This system is helpful to well understand some nonlinear PDEs and other nonlinear problems. Different from the case of $\alpha=\beta,\gamma=\tau$, it is more difficult to handle the critical condition. Fortunately, by applying the special iteration scheme and some critical asymptotic analysis, we establish the sharp criteria for existence and nonexistence of positive solutions to system (0.1). Then, we use the method of moving planes to prove the symmetry and monotonicity for the positive solutions of (0.1) when $c_{1}(x)\equiv c_{2}(x)\equiv1$ in the case \begin{eqnarray} \frac{\gamma-1}{p+\gamma-1}+\frac{\tau-1}{q+\tau-1}=\frac{n-\alpha\tau}{2n-\alpha\tau+\beta\gamma} +\frac{n-\beta\gamma}{2n-\beta\gamma+\alpha\tau}. \end{eqnarray}
    Mathematics Subject Classification: Primary: 35J50, 45G15, 45M20.


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