February  2016, 36(2): 601-609. doi: 10.3934/dcds.2016.36.601

A priori estimates for semistable solutions of semilinear elliptic equations

1. 

ICREA and Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada 1, Diagonal 647, 08028 Barcelona

2. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

3. 

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218

Received  June 2014 Revised  February 2015 Published  August 2015

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 $f'$.
Citation: Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
References:
[1]

H. Brezis, Is there failure of the Inverse Function Theorem?,, Morse Theory, 1 (2003), 23.   Google Scholar

[2]

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

[3]

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

[4]

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

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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.  doi: 10.1007/BF00280741.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

show all references

References:
[1]

H. Brezis, Is there failure of the Inverse Function Theorem?,, Morse Theory, 1 (2003), 23.   Google Scholar

[2]

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

[3]

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

[4]

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

[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.  doi: 10.1007/BF00280741.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

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