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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

show all references

References:
[1]

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

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[1]

Xavier Cabré, Manel Sanchón. Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian. Communications on Pure & Applied Analysis, 2007, 6 (1) : 43-67. doi: 10.3934/cpaa.2007.6.43

[2]

Asadollah Aghajani. Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3521-3530. doi: 10.3934/dcds.2017150

[3]

Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631

[4]

Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029

[5]

Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293

[6]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801

[7]

Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886

[8]

David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335

[9]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[10]

Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50

[11]

Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044

[12]

Soohyun Bae. Classification of positive solutions of semilinear elliptic equations with Hardy term. Conference Publications, 2013, 2013 (special) : 31-39. doi: 10.3934/proc.2013.2013.31

[13]

Alan V. Lair, Ahmed Mohammed. Entire large solutions of semilinear elliptic equations of mixed type. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1607-1618. doi: 10.3934/cpaa.2009.8.1607

[14]

Zhijun Zhang. Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1381-1392. doi: 10.3934/cpaa.2013.12.1381

[15]

Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41

[16]

Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733

[17]

Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943

[18]

Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561

[19]

Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033

[20]

Xavier Cabré. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 331-359. doi: 10.3934/dcds.2002.8.331

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]