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

Previous Article
Sharp large time behaviour in $ N $-dimensional Fisher-KPP equations
DCDS Home
This Issue
Next Article
Free boundaries subject to topological constraints

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 and 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.
[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.
[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.
[4]	H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
[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.
[6]	X. Cabré, Boundedness of stable solutions to semilinear elliptic equations: A survey, Adv. Nonlinear Stud., 17 (2017), 355-368.
[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.
[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.
[9]	X. Cabré and P. Miraglio, Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, forthcoming.
[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.
[11]	X. Cabré and T. Sanz-Perela, BMO and $L^\infty$ estimates for stable solutions to fractional semilinear elliptic equations, forthcoming.
[12]	G. Carron, Inégalités de Hardy sur les variétés Riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), 883-891.
[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.
[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.
[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.
[16]	P. Miraglio, Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian, preprint arXiv: 1907.13027.
[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.
[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.
[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.
[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.
[21]	S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133. doi: 10.1016/j.aim.2012.11.015.

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.
[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.
[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.
[4]	H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
[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.
[6]	X. Cabré, Boundedness of stable solutions to semilinear elliptic equations: A survey, Adv. Nonlinear Stud., 17 (2017), 355-368.
[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.
[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.
[9]	X. Cabré and P. Miraglio, Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, forthcoming.
[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.
[11]	X. Cabré and T. Sanz-Perela, BMO and $L^\infty$ estimates for stable solutions to fractional semilinear elliptic equations, forthcoming.
[12]	G. Carron, Inégalités de Hardy sur les variétés Riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), 883-891.
[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.
[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.
[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.
[16]	P. Miraglio, Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian, preprint arXiv: 1907.13027.
[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.
[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.
[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.
[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.
[21]	S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133. doi: 10.1016/j.aim.2012.11.015.

[1]	Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Xavier Cabré
on Google Scholar
Xavier Cabré

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Xavier Cabré
on Google Scholar
Xavier Cabré

[Back to Top]