\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A priori estimates for semistable solutions of semilinear elliptic equations

Abstract Related Papers Cited by
  • We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\leq 9$, but only established for $n\leq 4$.
        In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound will follow from a $L^p$-estimate for $f'(u)$ for every $p<3$ (and for all $n\geq 2$). Under a similar but more restrictive assumption on $f$, we also prove the $L^\infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$.
    Mathematics Subject Classification: Primary: 35K57, 35B65.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    H. Brezis, Is there failure of the Inverse Function Theorem?, Morse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations Int. Press, Somerville, 1 (2003), 23-33.

    [2]

    H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.

    [3]

    X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380.doi: 10.1002/cpa.20327.

    [4]

    X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733.doi: 10.1016/j.jfa.2005.12.018.

    [5]

    M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), 207-218.doi: 10.1007/BF00280741.

    [6]

    G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris , 330 (2000), 997-1002.doi: 10.1016/S0764-4442(00)00289-5.

    [7]

    M. Sanchón, Boundedness of the extremal solution of some $p$-Laplacian problems, Nonlinear Analysis, 67 (2007), 281-294.doi: 10.1016/j.na.2006.05.010.

    [8]

    J. Serrin, Local behavior of solutions of quasilinear elliptic equations, Acta Math., 111 (1964), 247-302.doi: 10.1007/BF02391014.

    [9]

    N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa. (3), 27 (1973), 265-308.

    [10]

    S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.doi: 10.1016/j.aim.2012.11.015.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(162) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return