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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
Article
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Abstract

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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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.

References

  1. [1]
    M. F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorbtion terms, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999), 229–271.MathSciNetzbMATHGoogle Scholar
  2. [2]
    M. F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Analyse Math. 84 (2001), 1–49.MathSciNetCrossRefGoogle Scholar
  3. [3]
    M. F. Bidaut-Véron and C. Yarur, Semilinear elliptic equations and systems with meaure data: Existence and a priori estimates, Adv. Differential Equations 7 (2002), 257–296.MathSciNetzbMATHGoogle Scholar
  4. [4]
    H. Brezis and P-L. Lions, A note on isolated singularities for linear elliptic equations, in Mathematical Analysis and Applications, Part A, Academic Press, New York-London, 1981, pp. 263–266.Google Scholar
  5. [5]
    H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of − Δu = V(x)e uin two dimensions, Comm. Partial Differential Equations 16 (1991), 1223–1253.MathSciNetCrossRefGoogle Scholar
  6. [6]
    D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.CrossRefGoogle Scholar
  7. [7]
    V. Maz’ya, Sobolev Spaces, with Applications to Elliptic Partial Differential Equations, Springer, Berlin, 2011.zbMATHGoogle Scholar
  8. [8]
    E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993), 125–151.MathSciNetCrossRefGoogle Scholar
  9. [9]
    E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems inN, Differential Integral Equations 9 (1996), 465–479.MathSciNetzbMATHGoogle Scholar
  10. [10]
    E. Mitidieri and S. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001), 1–384MathSciNetzbMATHGoogle Scholar
  11. [11]
    N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math. (2) 168 (2008), 859–914.MathSciNetCrossRefGoogle Scholar
  12. [12]
    P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic systems, Duke Math. J. 139 (2007), 555–579.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math. 221 (2009), 1409–1427.MathSciNetCrossRefGoogle Scholar
  14. [14]
    S. D. Taliaferro, Isolated singularities of nonlinear elliptic inequalities, Indiana Univ. Math. J. 50 (2001), 1885–1897.MathSciNetCrossRefGoogle Scholar
  15. [15]
    S. D. Taliaferro, Isolated singularities of nonlinear elliptic inequalities. II. Asymptotic behavior of solutions, Indiana Univ. Math. J. 55 (2006), 1791–1812.MathSciNetCrossRefGoogle Scholar
  16. [16]
    S. D. Taliaferro, Pointwise bounds and blow-up for nonlinear polyharmonic inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), 1069–1096.MathSciNetCrossRefGoogle Scholar

Copyright information

© 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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