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Non Degeneracy of Critical Points of the Robin Function with Respect to Deformations of the Domain

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Abstract

We show a result of genericity for non degenerate critical points of the Robin function with respect to deformations of the domain.

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References

  1. Bahri, A., Li, Y., Rey, O.: On a variational problem with lack of compactness: the topological effect of the critical points at infinity. Calc. Var. Partial Differ. Equ. 3(1), 67–93 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bandle, C., Flucher, M.: Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s equations − ΔU = e U and − ΔU = U (n + 2)/(n − 2). SIAM Rev. 38(2), 191–238 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baraket, S., Pacard, F.: Construction of singular limits for a semilinear elliptic equation in dimension 2. Calc. Var. Partial Differ. Equ. 6(1), 1–38 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Caffarelli, L.A., Friedman, A.: Convexity of solutions of semilinear elliptic equations. Duke Math. J. 52(2), 431–456 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  5. del Pino, M., Kowalczyk, M., Musso, M.: Singular limits in Liouville-type equations. Calc. Var. Partial Differ. Equ. 24(1), 47–81 (2005)

    Article  MATH  Google Scholar 

  6. Esposito, P., Grossi, M., Pistoia, A.: On the existence of blowing-up solutions for a mean field equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(2), 227–257 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ge, Y., Jing, R., Pacard, F.: Bubble towers for supercritical semilinear elliptic equations. J. Funct. Anal. 221(2), 251–302 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. In: Grundlehren der Mathematischen Wissenschaften, vol. 224, x+401 pp. Springer, Berlin New York (1977)

    Google Scholar 

  9. Gladiali, F., Grossi, M.: Some results for the Gelfand’s problem. Commun. Partial Differ. Equ. 29(9–10), 1335–1364 (2004)

    MATH  MathSciNet  Google Scholar 

  10. Grossi, M.: On the nondegeneracy of the critical points of the Robin function in symmetric domains. C. R. Math. Acad. Sci. Paris 335(2), 157–160 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ma, L., Wei, J.: Convergence for a Liouville equation. Comment. Math. Helv. 76(3), 506–514 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Pistoia, A.: On the uniqueness of solutions for a semilinear elliptic problem in convex domains. Differ. Integr. Equ. 17(11–12), 1201–1212 (2004)

    MATH  MathSciNet  Google Scholar 

  13. Quinn, F.: Transversal approximation on Banach manifolds. In: 1970 Global Analysis (Proc. Sympos. Pure Math., vol. XV, Berkeley, California), pp. 213–222. Amer. Math. Soc., Providence, R.I. (1968)

    Google Scholar 

  14. Rey, O.: The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89(1), 1–52 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. Saut, J.-C., Temam, R.: Generic properties of nonlinear boundary value problems. Partial Differ. Equ. 4(3), 293–319 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  16. Uhlenbeck, K.: Generic properties of eigenfunctions. Am. J. Math. 98(4), 1059–1078 (1976)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Angela Pistoia.

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Micheletti, A.M., Pistoia, A. Non Degeneracy of Critical Points of the Robin Function with Respect to Deformations of the Domain. Potential Anal 40, 103–116 (2014). https://doi.org/10.1007/s11118-013-9340-2

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  • DOI: https://doi.org/10.1007/s11118-013-9340-2

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