Existence and blow-up behavior of positive solutions to integral equations on bounded domains.

Consider the integral equation$ \begin{eqnarray*} f^{m-1}(x)+f^{q-1}(x) = \int_{\Omega}^{}\frac{f(y)}{\left | x-y \right |^{n-\alpha } }dy, \ \ \ \end{eqnarray*} $where $ 1<\alpha<n $, $ \Omega\subset{\mathbb{R}}^{n} $ is a smooth bounded domain. The existence of energy maximizing positive solutions in the subcritical case $ 2>q,m >\frac{2n}{n+\alpha} $, and the nonexistence of energy maximizing positive solutions in the critical and supercritical case $ 1<q\le \frac{2n}{n+\alpha}, 1<m\le \frac{2n}{n+\alpha} $ are obtained. Based on these, the blow-up behavior of energy maximizing positive solutions as $ (q,m)\to (q_\alpha, q_\alpha) $ are studied. [ABSTRACT FROM AUTHOR]