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manuscripta mathematica

, Volume 154, Issue 3–4, pp 345–357 | Cite as

A note on the local regularity of distributional solutions and subsolutions of semilinear elliptic systems

  • Rainer Mandel
Article
  • 86 Downloads

Abstract

In this note we prove local regularity results for distributional solutions and subsolutions of semilinear elliptic systems such as
$$\begin{aligned} L_k^m u_k = f_k(x,u_1,\ldots ,u_N) \quad \text {in }\mathbb {R}^n\qquad (k=1,\ldots ,N \text { and }m\in \mathbb {N}) \end{aligned}$$
where \(L_1,\ldots ,L_N\) are of divergence-form and \(n\ge 2m\). We show that distributional subsolutions are locally bounded from above if \(f_k(x,z)\le C(1+|z|^p)\) for \(1\le p<\frac{n}{n-2m}\) and \(k=1,\ldots ,N\). Furthermore, regularity properties of solutions and improved versions for bounded subsolutions are given. Even for \(f_1=\ldots =f_N=0\) our results are new.

Mathematics Subject Classification

35A23 35B65 35J48 35J61 

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References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Volume 140 of Pure and Applied Mathematics (Amsterdam), 2nd edn. Elsevier, Amsterdam (2003)Google Scholar
  2. 2.
    Amann, H., Quittner, P.: Elliptic boundary value problems involving measures: existence, regularity, and multiplicity. Adv. Differ. Equ. 3(6), 753–813 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bao, J., Zhang, W.: Regularity of very weak solutions for elliptic equation of divergence form. J. Funct. Anal. 262(4), 1867–1878 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bao, J., Zhang, W.: Regularity of very weak solutions for nonhomogeneous elliptic equation. Commun. Contemp. Math. 15(4), 1350012 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brezis, B.: On a conjecture of J. Serrin. Rend. Lincei Math. Appl. 19(4), 335–338 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cassani, D., do Ó, J.M., Ghoussoub, N.: On a fourth order elliptic problem with a singular nonlinearity. Adv. Nonlinear Stud. 9(1), 177–197 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dolzmann, G., Müller, S.: Estimates for Green’s matrices of elliptic systems by \(L^p\) theory. Manuscr. Math. 88(2), 261–273 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Duzaar, F., Mingione, G.: Gradient estimates via non-linear potentials. Am. J. Math. 133(4), 1093–1149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gazzola, F., Grunau, H.-C., Sweers, G.: Polyharmonic boundary value problems, volume 1991 of Lecture Notes in Mathematics. Springer, Berlin (2010). Positivity preserving and nonlinear higher order elliptic equations in bounded domainsGoogle Scholar
  10. 10.
    Gilbarg, D., Trudinger, N.S: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 editionGoogle Scholar
  11. 11.
    Grüter, M., Widman, K.-O.: The Green function for uniformly elliptic equations. Manuscr. Math. 37(3), 303–342 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hager, R.A., Ross, J.: A regularity theorem for second order elliptic divergence equations. Ann. Sc. Norm. Super. Cl. Sci. 26(2), 283–290 (1972)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Jin, T., Maz’ya, V., Van Schaftingen, J.: Pathological solutions to elliptic problems in divergence form with continuous coefficients. C. R. Math. Acad. Sci. Paris 347(13–14), 773–778 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lieb, E.H., Loss, M.: Analysis, volume 14 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence, RI (2001)Google Scholar
  15. 15.
    Mandel, R., Reichel, W.: Distributional solutions of the stationary nonlinear Schrödinger equation: singularities, regularity and exponential decay. Z. Anal. Anwend. 32(1), 55–82 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mazzeo, R., Pacard, F.: A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis. J. Differ. Geom. 44(2), 331–370 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mingione, G.: The Calderón–Zygmund theory for elliptic problems with measure data. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6(2), 195–261 (2007)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mitrea, D.: Distributions, partial differential equations, and harmonic analysis. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  19. 19.
    Pacard, F.: Existence de solutions faibles positives de \(-\Delta u=u^\alpha \) dans des ouverts bornés de \({ R}^n\), \(n\ge 3\). C. R. Acad. Sci. Paris Sér. I Math. 315(7), 793–798 (1992)MathSciNetGoogle Scholar
  20. 20.
    Pacard, F.: Existence and convergence of positive weak solutions of \(-\Delta u=u^{n/(n-2)}\) in bounded domains of \(\mathbf{R}^n, n\ge 3\). Calc. Var. Partial Differ. Equ. 1(3), 243–265 (1993)CrossRefzbMATHGoogle Scholar
  21. 21.
    Rébaï, Y.: Weak solutions of the problem \(\Delta ^2u=u^{\frac{n}{n-4}}\) with prescribed singular sets. Adv. Nonlinear Stud. 8(4), 719–744 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Serrin, J.: Pathological solutions of elliptic differential equations. Ann. Sc. Norm. Super. Pisa 3(18), 385–387 (1964)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Karlsruher Institut für TechnologieInstitut für AnalysisKarlsruheGermany

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