Regularity of solutions to an integral equation associated with Bessel potential (Q652022)

The authors study the differential equation in \(\mathbb{R}^n\) \[ (I-\Delta)^{\frac{\alpha}{2}} u = u^\beta,\quad \alpha >0, \;\beta > 1. \] They prove that if \(u\) is a positive solution of this equation in the space \(L^q (\mathbb{R}^n), \;q > \max (\beta, \frac{n(\beta - 1)}{\alpha}),\) then \(u\) is a bounded and Lipschitz continuous function. There is a survey of previous results in the article.