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

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.

Citation: Xavier Cabré. A new proof of the boundedness results for stable solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 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. BrezisT. CazenaveY. 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. BrezisT. CazenaveY. 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

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