# American Institute of Mathematical Sciences

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