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Asymptotic behavior at isolated singularities for solutions of nonlocal semilinear elliptic systems of inequalities

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Abstract

We study the behavior near the origin of \(C^2\) positive solutions \(u(x)\) and \(v(x)\) of the system

$$\begin{aligned} \left\{ \begin{aligned} 0\le -\Delta u\le \left( \frac{1}{|x|^\alpha }* v\right) ^\lambda \\ 0\le -\Delta v\le \left( \frac{1}{|x|^\beta }* u\right) ^\sigma \end{aligned} \quad \text{ in } B_2(0){\setminus }\{0\}\subset {\mathbb R}^n, n\ge 3, \right. \end{aligned}$$

where \(\lambda ,\sigma \ge 0\) and \(\alpha , \beta \in (0,n)\). A by-product of our methods used to study these solutions will be results on the behavior near the origin of \(L^1(B_1(0))\) solutions \(f\) and \(g\) of the system

$$\begin{aligned} \left\{ \begin{aligned} 0\le f(x)\le C\left( |x|^{2-\alpha } +\int _{|y|<1}\frac{ g(y)\,dy}{|x-y|^{\alpha -2}} \right) ^\lambda \\ 0\le g(x)\le C\left( |x|^{2-\beta } +\int _{|y|<1}\frac{ f(y)\,dy}{|x-y|^{\beta -2}} \right) ^\sigma \end{aligned}\right. \qquad \text{ for } 0<|x|<1 \end{aligned}$$

where \(\lambda ,\sigma \ge 0\) and \(\alpha , \beta \in (2,n+2)\).

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Acknowledgments

The authors would like to thank Stephen J. Gardiner for helpful discussions.

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Correspondence to Steven D. Taliaferro.

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Communicated by P. Rabinowitz.

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Ghergu, M., Taliaferro, S.D. Asymptotic behavior at isolated singularities for solutions of nonlocal semilinear elliptic systems of inequalities. Calc. Var. 54, 1243–1273 (2015). https://doi.org/10.1007/s00526-015-0824-3

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