A priori estimates for semistable solutions of semilinear elliptic equations

Xavier Cabré; Manel  Sanchón; Joel Spruck

doi:10.3934/dcds.2016.36.601

Article Contents

2016, Volume 36, Issue 2: 601-609. Doi: 10.3934/dcds.2016.36.601

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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
3.
Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218

Received: June 2014

Revised: February 2015

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

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 $f'$.

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Mathematics Subject Classification: Primary: 35K57, 35B65.

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References

[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.