# Pointwise bounds and blow-up for systems of semilinear elliptic inequalities at an isolated singularity via nonlinear potential estimates

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## Abstract

We study the behavior near the origin of where

*C*^{2}positive solutions*u*(*x*) and*v*(*x*) of the system$$\matrix{{0 \le - {\rm{\Delta}}u \le f(v)} \\ {0 \le - {\rm{\Delta}}v \le g(u)} \\} \quad {\rm{in}}\,{B_1}\left(0 \right)\,\backslash \left\{0\right\}\, \subset {\mathbb{R}^n},\,n \ge 2,$$

*f, g*:(0, ∞) → (0, ∞) are continuous functions. We provide optimal conditions on*f*and*g*at ∞ such that solutions of this system satisfy pointwise bounds near the origin. In dimension*n*= 2 we show that this property holds if log^{+}*f*or log^{+}*g*grow at most linearly at infinity. In dimension*n*≥ 3 and under the assumption*f*(*t*) =*O*(*t*^{λ}),*g*(*t*) =*O*(*t*^{σ})*as t*→ ∞ (*λ, σ*≥ 0), we obtain a new critical curve that optimally describes the existence of such pointwise bounds. Our approach relies in part on sharp estimates of nonlinear potentials which appear naturally in this context.## Preview

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## Notes

### Acknowledgement

The authors would like to thank Stephen J. Gardiner for helpful discussions as well as the anonymous referee for her/his pertinent suggestions which led to an improvement of our presentation.

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