Wolff-type integral system including \(m\) equations. (English) Zbl 1492.45005
The following Wolff-type integral system is considered:
\[
\begin{cases}
u_i(x)=C_i(x)W_{\beta,\gamma}(u_{i+1}^{p_{i+1}})(x), \quad i=1,\ldots,m-1, \\
u_m(x)=C_m(x)W_{\beta,\gamma}(u_{1}^{p_{1}})(x),
\end{cases}
\]
where \(n\geq 1\), \(p_i\neq 0\), \(\gamma>1\), \(\beta>0\) and \(\beta\gamma\neq n\). The double-bounded functions \(C_(x)\) satisfy the conditions
\[
0<c\leq C_i(x)\leq C<\infty, \quad i=1,\dots,m.
\]
Here, \(W_{\beta,\gamma}(f)(x)\) is the Wolff potential of a positive function \(f\).
According to the different forms of Wolff-type integral system, the authors discuss the properties of the positive solutions in two cases: \(p_i<0\) and \(p_i>0\). A special iteration scheme in integral form (as well as in differential form) is applied. Using the Wolff potential integral estimates the asymptotic rates and the integrability of positive solutions are obtained.
According to the different forms of Wolff-type integral system, the authors discuss the properties of the positive solutions in two cases: \(p_i<0\) and \(p_i>0\). A special iteration scheme in integral form (as well as in differential form) is applied. Using the Wolff potential integral estimates the asymptotic rates and the integrability of positive solutions are obtained.
Reviewer: Alexander N. Tynda (Penza)
MSC:
| 45G15 | Systems of nonlinear integral equations |
| 45M20 | Positive solutions of integral equations |
| 45M05 | Asymptotics of solutions to integral equations |
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