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

  • Marius Ghergu
  • Steven D. TaliaferroEmail author
  • Igor E. Verbitsky


We study the behavior near the origin of C2 positive solutions u(x) and v (x) of the system
$$\matrix{{0 \le - {\rm{\Delta}}u \le f(v)} \\ {0 \le - {\rm{\Delta}}v \le g(u)} \\} \quad {\rm{in}}\,{B_1}\left(0 \right)\,\backslash \left\{0\right\}\, \subset {\mathbb{R}^n},\,n \ge 2,$$
where 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(tλ), g(t) = O(tσ)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.


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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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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Marius Ghergu
    • 1
  • Steven D. Taliaferro
    • 2
    Email author
  • Igor E. Verbitsky
    • 3
  1. 1.School of Mathematics and StatisticsUniversity College DublinBelfield, Dublin 4Ireland
  2. 2.Mathematics DepartmentTexas A&M UniversityCollege StationUSA
  3. 3.Department of MathematicsUniversity of MissouriColumbiaUSA

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