American Institute of Mathematical Sciences

Advanced
December  2019, 39(12): 7249-7264. doi: 10.3934/dcds.2019302

A new proof of the boundedness results for stable solutions to semilinear elliptic equations

Xavier Cabré 1,2,3,

1. 

ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain
2. 

Universitat Politècnica de Catalunya, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain
3. 

BGSMath, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Spain

Dedicated to Luis Caffarelli, with friendship and great admiration

Received  February 2019 Revised  May 2019 Published  September 2019

Fund Project: Xavier Cabré is supported by grants MTM2017-84214-C2-1-P and MdM-2014-0445 (Government of Spain), and is a member of the research group 2017SGR1392 (Government of Catalonia).

We consider the class of stable solutions to semilinear equations $ -\Delta u = f(u) $ in a bounded smooth domain of $ \mathbb{R}^n $. Since 2010 an interior a priori $ L^\infty $ bound for stable solutions is known to hold in dimensions $ n\le 4 $ for all $ C^1 $ nonlinearities $ f $. In the radial case, the same is true for $ n\leq 9 $. Here we provide with a new, simpler, and unified proof of these results. It establishes, in addition, some new estimates in higher dimensions —for instance $ L^p $ bounds for every finite $ p $ in dimension 5.

Since the mid nineties, the existence of an $ L^\infty $ bound holding for all $ C^1 $ nonlinearities when $ 5\leq n\leq 9 $ was a challenging open problem. This has been recently solved by A. Figalli, X. Ros-Oton, J. Serra, and the author, for nonnegative nonlinearities, in a forthcoming paper.

Keywords: Stable solutions, semilinear elliptic equations, boundedness of solutions.
Mathematics Subject Classification: 35K57, 35B65.
Citation: Xavier Cabré. A new proof of the boundedness results for stable solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7249-7264. doi: 10.3934/dcds.2019302
References:
[1]

D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778.  doi: 10.1215/S0012-7094-75-04265-9.  Google Scholar

[2]

H. Brezis, Is there failure of the Inverse Function Theorem?, in Morse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations, New Stud. Adv. Math., 1, Int. Press, Somerville, MA, 1 (2003), 23–33.  Google Scholar

[3]

H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.   Google Scholar

[4]

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

[5]

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

[6]

X. Cabré, Boundedness of stable solutions to semilinear elliptic equations: A survey, Adv. Nonlinear Stud., 17 (2017), 355-368.   Google Scholar

[7]

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

[8]

X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, preprint arXiv: 1907.09403. Google Scholar

[9]

X. Cabré and P. Miraglio, Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, forthcoming. Google Scholar

[10]

X. Cabré and G. Poggesi, Stable solutions to some elliptic problems: Minimal cones, the Allen-Cahn equation, and blow-up solutions, Geometry of PDEs and Related Problems, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Cham, 2220 (2018), 1–45.  Google Scholar

[11]

X. Cabré and T. Sanz-Perela, BMO and $L^\infty$ estimates for stable solutions to fractional semilinear elliptic equations, forthcoming. Google Scholar

[12]

G. Carron, Inégalités de Hardy sur les variétés Riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), 883-891.   Google Scholar

[13]

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

[14]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 143, Boca Raton, FL, 2011. doi: 10.1201/b10802.  Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition. Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[16]

P. Miraglio, Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian, preprint arXiv: 1907.13027. Google Scholar

[17]

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

[18]

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

[19]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.  doi: 10.1007/s002050050081.  Google Scholar

[20]

P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.   Google Scholar

[21]

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

show all references

References:
[1]

D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778.  doi: 10.1215/S0012-7094-75-04265-9.  Google Scholar

[2]

H. Brezis, Is there failure of the Inverse Function Theorem?, in Morse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations, New Stud. Adv. Math., 1, Int. Press, Somerville, MA, 1 (2003), 23–33.  Google Scholar

[3]

H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.   Google Scholar

[4]

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

[5]

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

[6]

X. Cabré, Boundedness of stable solutions to semilinear elliptic equations: A survey, Adv. Nonlinear Stud., 17 (2017), 355-368.   Google Scholar

[7]

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

[8]

X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, preprint arXiv: 1907.09403. Google Scholar

[9]

X. Cabré and P. Miraglio, Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, forthcoming. Google Scholar

[10]

X. Cabré and G. Poggesi, Stable solutions to some elliptic problems: Minimal cones, the Allen-Cahn equation, and blow-up solutions, Geometry of PDEs and Related Problems, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Cham, 2220 (2018), 1–45.  Google Scholar

[11]

X. Cabré and T. Sanz-Perela, BMO and $L^\infty$ estimates for stable solutions to fractional semilinear elliptic equations, forthcoming. Google Scholar

[12]

G. Carron, Inégalités de Hardy sur les variétés Riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), 883-891.   Google Scholar

[13]

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

[14]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 143, Boca Raton, FL, 2011. doi: 10.1201/b10802.  Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition. Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[16]

P. Miraglio, Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian, preprint arXiv: 1907.13027. Google Scholar

[17]

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

[18]

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

[19]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.  doi: 10.1007/s002050050081.  Google Scholar

[20]

P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.   Google Scholar

[21]

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

[1]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[2]

Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 3805-3816. doi: 10.3934/era.2021062
[3]

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
[4]

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
[5]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801
[6]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
[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]

Massimo Grossi. On the number of critical points of solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 4215-4228. doi: 10.3934/era.2021080
[10]

Mostafa Fazly, Yuan Li. Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4185-4206. doi: 10.3934/dcds.2021033
[11]

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
[12]

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
[13]

Jinlong Bai, Desheng Li, Chunqiu Li. A note on multiplicity of solutions near resonance of semilinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3351-3365. doi: 10.3934/cpaa.2019151
[14]

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
[15]

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
[16]

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
[17]

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
[18]

Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166
[19]

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
[20]

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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (294)
  • HTML views (187)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy