Geodesic
Global pointwise estimates of positive solutions to sublinear equations
Algebra i analiz, Tome 34 (2022) no. 3, pp. 296-330

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Bilateral pointwise estimates are provided for positive solutions $u$ to the sublinear integral equation $$ u = \mathbf{G}(\sigma u^q) + f \textrm{ in } \Omega, $$ for $0 q 1$, where $\sigma\ge 0$ is a measurable function or a Radon measure, $ f \ge 0$, and $\mathbf{G}$ is the integral operator associated with a positive kernel $G$ on $\Omega\times\Omega$. The main results, which include the existence criteria and uniqueness of solutions, hold true for quasi-metric, or quasi-metrically modifiable kernels $G$. As a consequence, bilateral estimates, are obtained, along with existence and uniqueness, for positive solutions $u$, possibly unbounded, to sublinear elliptic equations involving the fractional Laplacian, $$ (-\Delta)^{\frac{\alpha}{2}} u = \sigma u^q + \mu \textrm{ in } \Omega, u=0 \textrm{ in } \Omega^c, $$ where $0$, and $\mu, \sigma \ge 0$ are measurable functions, or Radon measures, on a bounded uniform domain $\Omega \subset \mathbb{R}^n$ for $0 \alpha \le 2$, or on the entire space $\mathbb{R}^n$, a ball or half-space, for $0 \alpha $.
Keywords: sublinear equations, Green's kernel, weak maximum principle.
Mots-clés : quasi-metric kernels 
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I. E. Verbitsky. Global pointwise estimates of positive solutions to sublinear equations. Algebra i analiz, Tome 34 (2022) no. 3, pp. 296-330. http://geodesic.mathdoc.fr/item/AA_2022_34_3_a13/

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