Rocky Mountain Journal of Mathematics

Qualitative properties and standard estimates of solutions for some fourth order elliptic equations

Kaisheng Liu and Ruichang Pei

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In this paper, first, we make the estimates for a class of fourth order elliptic equations in different domains and boundary conditions. Consequently, we study the qualitative properties of solutions with prescribed $Q$-curvature. Finally, we also will obtain some radially symmetric results by using moving plane methods.

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Rocky Mountain J. Math., Volume 44, Number 3 (2014), 975-986.

First available in Project Euclid: 28 September 2014

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Elliptic equations of fourth order asymptotic behavior a-priori estimates Q curvature


Liu, Kaisheng; Pei, Ruichang. Qualitative properties and standard estimates of solutions for some fourth order elliptic equations. Rocky Mountain J. Math. 44 (2014), no. 3, 975--986. doi:10.1216/RMJ-2014-44-3-975.

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  • H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\triangle u=V(x)e^{u}$ in two dimensions, Comm. PDE 16 (1991), 1223–1253.
  • W.X. Chen and C.M. Li, Classification of solutions of some nonlinear elliptic equation, Duke Math. J. 63 (1991), 615–622.
  • ––––, Qualitative properties of solutions to some nonlinear elliptic equations in $R^2$, Duke Math. J. 71 (1993), 427–439.
  • D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, second ed., Grundl. Math. Wissen. 224, Springer, Berlin, 1983.
  • C.S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv. 73 (1998), 206–231.
  • L. Martinazzi, Classification of solutions to the higher order Liouville's equation on $R^{2m}$, Math. Z. 263 (2009), 307–329.
  • J.C. Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Comm. PDE 21 (1996), 1451–1467.
  • J.C. Wei and X.W. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (1999), 207–228.
  • ––––, Prescribing $Q$-curvature problem on $S^n$, J. Funct. Anal. 257 (2009), 1995–2023.
  • X.W. Xu, Classification of solutions of certain fourth-order nonlinear elliptic equations in $R^4$, Pacific J. Math. 225 (2006), 361–378.
  • X.W. Xu, Uniqueness and non-existence theorems for conformally invariant equations, J. Funct. Anal. 222 (2005), 1–28.