# American Institute of Mathematical Sciences

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, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
##### References:
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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.  Google Scholar [2] H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.  Google Scholar [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.  Google Scholar [4] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733. 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), 207-218. 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), 997-1002. 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), 281-294. 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), 247-302. doi: 10.1007/BF02391014.  Google Scholar [9] N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa. (3), 27 (1973), 265-308. Google Scholar [10] S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133. doi: 10.1016/j.aim.2012.11.015.  Google Scholar
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