Recent progress on stable and finite Morse index solutions of semilinear elliptic equations

Kelei Wang; Kelei Wang

doi:10.3934/era.2021062

Electronic Research Archive

2021, Volume 29, Issue 6: 3805-3816. doi: 10.3934/era.2021062

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Recent progress on stable and finite Morse index solutions of semilinear elliptic equations

Kelei Wang

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received: 01 June 2021 Published: 13 August 2021
Primary: 35J61, 35B35; Secondary: 35B08

We discuss some recent results (mostly from the last decade) on stable and finite Morse index solutions of semilinear elliptic equations, where Norman Dancer has made many important contributions. Some open questions in this direction are also discussed.
- Stable solutions,
- finite Morse index solutions,
- Liouville theorem,
- De Giorgi conjecture
Citation: Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations[J]. Electronic Research Archive, 2021, 29(6): 3805-3816. doi: 10.3934/era.2021062

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Abstract

We discuss some recent results (mostly from the last decade) on stable and finite Morse index solutions of semilinear elliptic equations, where Norman Dancer has made many important contributions. Some open questions in this direction are also discussed.

References

[1]	Higher-dimensional catenoid, Liouville equation, and Allen-Cahn equation. Int. Math. Res. Not. IMRN (2016) 2016: 7051-7102.
[2]	Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. (1981) 325: 105-144.
[3]	A free boundary problem for semilinear elliptic equations. J. Reine Angew. Math. (1986) 368: 63-107.
[4]	Entire solutions of semilinear elliptic equations in $\mathbb{R}^3$ and a conjecture of De Giorgi. J. Amer. Math. Soc. (2000) 13: 725-739.
[5]	H. Brezis, Is there failure of the inverse function theorem?, In Morse theory, minimax theory and their applications to nonlinear differential equations, volume 1 of New Stud. Adv. Math., pages 23–33. Int. Press, Somerville, MA, (2003).
[6]	Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s\sim 1$. J. Reine Angew. Math. (2020) 764: 157-180.
[7]	Stable solutions to semilinear elliptic equations are smooth up to dimension 9. Acta Math. (2020) 224: 187-252.
[8]	Nonlocal minimal surfaces. Comm. Pure Appl. Math. (2010) 63: 1111-1144.
[9]	Regularity properties of nonlocal minimal surfaces via limiting arguments. Adv. Math. (2013) 248: 843-871.
[10]	L. A. Caffarelli, D. Jerison and C. E. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, In Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, volume 350 of Contemp. Math., pages 83–97. Amer. Math. Soc., Providence, RI, (2004). doi: 10.1090/conm/350/06339
[11]	The structure of stable minimal hypersurfaces in $\mathbb{R}^{n+1}$. Math. Res. Lett. (1997) 4: 637-644.
[12]	Minimal surfaces and the Allen-Cahn equation on 3-manifolds: Index, multiplicity, and curvature estimates. Ann. of Math. (2) (2020) 191: 213-328.
[13]	Stable and finite Morse index solutions on $\mathbb{R}^n$ or on bounded domains with small diffusion. II. Indiana Univ. Math. J. (2004) 53: 97-108.
[14]	Stable solutions on $\mathbb{R}^n$ and the primary branch of some non-self-adjoint convex problems. Differential Integral Equations (2004) 17: 961-970.
[15]	Stable and finite Morse index solutions on $\mathbb{R}^n$ or on bounded domains with small diffusion. Trans. Amer. Math. Soc. (2005) 357: 1225-1243.
[16]	Supercritical finite Morse index solutions. Nonlinear Anal. (2007) 66: 268-269.
[17]	Finite Morse index solutions of exponential problems. Ann. Inst. H. Poincaré Anal. Non Linéaire (2008) 25: 173-179.
[18]	Finite Morse index solutions of supercritical problems. J. Reine Angew. Math. (2008) 620: 213-233.
[19]	Stable and finite Morse index solutions for Dirichlet problems with small diffusion in a degenerate case and problems with infinite boundary values. Adv. Nonlinear Stud. (2009) 9: 657-678.
[20]	E. N. Dancer, Finite Morse index and linearized stable solutions on bounded and unbounded domains, In Proceedings of the International Congress of Mathematicians. Volume III, pages 1901–1909. Hindustan Book Agency, New Delhi, (2010).
[21]	New results for finite Morse index solutions on $\mathbb{R}^N$ and applications. Adv. Nonlinear Stud. (2010) 10: 581-595.
[22]	Finite Morse index solutions of an elliptic equation with supercritical exponent. J. Differential Equations (2011) 250: 3281-3310.
[23]	On the classification of solutions of $-\Delta u = e^u$ on $\mathbb{R}^N$: Stability outside a compact set and applications. Proc. Amer. Math. Soc. (2009) 137: 1333-1338.
[24]	Non-radial singular solutions of the Lane-Emden equation in $\mathbb{R}^N$. Indiana Univ. Math. J. (2012) 61: 1971-1996.
[25]	Partial regularity of finite Morse index solutions to the Lane-Emden equation. J. Funct. Anal. (2011) 261: 218-232.
[26]	A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem. Adv. Math. (2014) 258: 240-285.
[27]	On the fractional Lane-Emden equation. Trans. Amer. Math. Soc. (2017) 369: 6087-6104.
[28]	E. De Giorgi, Convergence problems for functionals and operators, In Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), pages 131–188. Pitagora, Bologna, (1979).
[29]	A singular energy minimizing free boundary. J. Reine Angew. Math. (2009) 635: 1-21.
[30]	Multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$. J. Funct. Anal. (2010) 258: 458-503.
[31]	The Toda system and clustering interfaces in the Allen-Cahn equation. Arch. Ration. Mech. Anal. (2008) 190: 141-187.
[32]	On De Giorgi's conjecture in dimension $N\geq 9$. Ann. of Math. (2) (2011) 174: 1485-1569.
[33]	Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$. J. Differential Geom. (2013) 93: 67-131.
[34]	Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature. Geom. Funct. Anal. (2010) 20: 918-957.
[35]	The work of Norman Dancer. Discrete Contin. Dyn. Syst. Ser. S (2019) 12: 1807-1833.
[36]	Positive solutions of an elliptic equation with negative exponent: Stability and critical power. J. Differential Equations (2009) 246: 2387-2414.
[37]	Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic equations. Adv. Differential Equations (2013) 18: 737-768.
[38]	Finite Morse index solutions of weighted elliptic equations and the critical exponents. Calc. Var. Partial Differential Equations (2015) 54: 3161-3181.
[39]	Monotonicity formula and $\varepsilon$-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions. Calc. Var. Partial Differential Equations (2014) 50: 615-638.
[40]	L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, volume 143 of Chapman $ & $ Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802
[41]	L. Dupaigne and A. Farina, Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains, arXiv: Analysis of PDEs, (2019).
[42]	P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, volume 20 of Courant Lecture Notes in Mathematics, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. doi: 10.1090/cln/020
[43]	On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$. J. Math. Pures Appl. (9) (2007) 87: 537-561.
[44]	Stable solutions of $-\Delta u = e^u$ on $\mathbb{R}^N$. C. R. Math. Acad. Sci. Paris (2007) 345: 63-66.
[45]	M. Fazly, Y. Hu and W. Yang, On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand-Liouville equation, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 11, 26 pp. doi: 10.1007/s00526-020-01874-7
[46]	M. Fazly and J. Wei, On stable solutions of the fractional Hénon-Lane-Emden equation, Commun. Contemp. Math., 18 (2016), 1650005, 24 pp. doi: 10.1142/S021919971650005X
[47]	On finite Morse index solutions of higher order fractional Lane-Emden equations. Amer. J. Math. (2017) 139: 433-460.
[48]	M. Fazly, J. Wei and W. Yang, Classification of finite Morse index solutions of higher-order Gelfand-Liouville equation, arXiv: Analysis of PDEs, (2020).
[49]	M. Fazly and W. Yang, On stable and finite Morse index solutions of the fractional Toda system, J. Funct. Anal., 280 (2021), 108870, 35 pp. doi: 10.1016/j.jfa.2020.108870
[50]	The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Amer. Math. Soc. (1970) 76: 767-771.
[51]	On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete Contin. Dyn. Syst. (2019) 39: 6945-6959.
[52]	On stable solutions for boundary reactions: A De Giorgi-type result in dimension $4+1$. Invent. Math. (2020) 219: 153-177.
[53]	On the oriented Plateau problem. Rend. Circ. Mat. Palermo (2) (1962) 11: 69-90.
[54]	The sharp exponent for a Liouville-type theorem for an elliptic inequality. Rend. Istit. Mat. Univ. Trieste (2003) 34: 99-102.
[55]	On a conjecture of De Giorgi and some related problems. Math. Ann. (1998) 311: 481-491.
[56]	C. Gui and Q. Li, Some energy estimates for stable solutions to fractional Allen-Cahn equations, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 49, 17 pp. doi: 10.1007/s00526-020-1701-2
[57]	Axially symmetric solutions of the Allen-Cahn equation with finite Morse index. Trans. Amer. Math. Soc. (2020) 373: 3649-3668.
[58]	On the symmetry of positive solutions of the Lane-Emden equation with supercritical exponent. Adv. Differential Equations (2002) 7: 641-666.
[59]	Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete Contin. Dyn. Syst. (2014) 34: 2561-2580.
[60]	Liouville-type theorems for the fourth order nonlinear elliptic equation. J. Differential Equations (2014) 256: 1817-1846.
[61]	A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system. Adv. Differential Equations (2017) 22: 49-76.
[62]	Stability for entire radial solutions to the biharmonic equation with negative exponents. C. R. Math. Acad. Sci. Paris (2018) 356: 632-636.
[63]	A. Hyder and W. Yang, Classification of stable solutions to a non-local Gelfand-Liouville equation, arXiv: Analysis of PDEs, (2020).
[64]	Higher critical points in an elliptic free boundary problem. J. Geom. Anal. (2018) 28: 1258-1294.
[65]	Some remarks on stability of cones for the one-phase free boundary problem. Geom. Funct. Anal. (2015) 25: 1240-1257.
[66]	Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. (1972/73) 49: 241-269.
[67]	The regularity and stability of solutions to semilinear fourth-order elliptic problems with negative exponents. Proc. Roy. Soc. Edinburgh Sect. A (2016) 146: 195-212.
[68]	Remarks on entire solutions for two fourth-order elliptic problems. Proc. Edinb. Math. Soc. (2) (2016) 59: 777-786.
[69]	Minimal hypersurfaces with finite index. Math. Res. Lett. (2002) 9: 95-103.
[70]	Senping Luo, Juncheng Wei and Wenming Zou, Monotonicity formula and classification of stable solutions to polyharmonic Lane-Emden equations, 2020.
[71]	Properties of positive solutions to an elliptic equation with negative exponent. J. Funct. Anal. (2008) 254: 1058-1087.
[72]	Stable and singular solutions of the equation $\Delta u = 1/u$. Indiana Univ. Math. J. (2004) 53: 1681-1703.
[73]	Liouville theorems for stable at infinity solutions of Lane-Emden system. Nonlinearity (2019) 32: 910-926.
[74]	Partial regularity for weak solutions of a nonlinear elliptic equation. Manuscripta Math. (1993) 79: 161-172.
[75]	Convergence and partial regularity for weak solutions of some nonlinear elliptic equation: The supercritical case. Ann. Inst. H. Poincaré Anal. Non Linéaire (1994) 11: 537-551.
[76]	Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones. J. Funct. Anal. (2013) 264: 1131-1167.
[77]	A minimization problem and the regularity of solutions in the presence of a free boundary. Indiana Univ. Math. J. (1983) 32: 1-17.
[78]	Regularity of flat level sets in phase transitions. Ann. of Math. (2) (2009) 169: 41-78.
[79]	R. Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, In Seminar on Minimal Submanifolds, volume 103 of Ann. of Math. Stud., pages 111–126. Princeton Univ. Press, Princeton, NJ, (1983).
[80]	Regularity of stable minimal hypersurfaces. Comm. Pure Appl. Math. (1981) 34: 741-797.
[81]	Curvature estimates for minimal hypersurfaces. Acta Math. (1975) 134: 275-288.
[82]	Minimal varieties in Riemannian manifolds. Ann. of Math. (2) (1968) 88: 62-105.
[83]	Topics of stable solutions to elliptic equations. Sugaku Expositions (2021) 34: 35-59.
[84]	Some Liouville theorems for Hénon type elliptic equations. J. Funct. Anal. (2012) 262: 1705-1727.
[85]	Partial regularity of stable solutions to the Emden equation. Calc. Var. Partial Differential Equations (2012) 44: 601-610.
[86]	Partial regularity of stable solutions to the supercritical equations and its applications. Nonlinear Anal. (2012) 75: 5238-5260.
[87]	Erratum to: Partial regularity of stable solutions to the Emden equation. Calc. Var. Partial Differential Equations (2013) 47: 433-435.
[88]	K. Wang, The structure of finite Morse index solutions of two phase transition models in $\mathbb{R}^2$, arXiv preprint, (2015), arXiv: 1506.00491.
[89]	Stable and finite Morse index solutions of Toda system. J. Differential Equations (2019) 268: 60-79.
[90]	Finite Morse index implies finite ends. Comm. Pure Appl. Math. (2019) 72: 1044-1119.
[91]	K. Wang and J. Wei, Second order estimate on transition layers, Adv. Math., 358 (2019), 106856, 85 pp. doi: 10.1016/j.aim.2019.106856
[92]	Partial regularity for a minimum problem with free boundary. J. Geom. Anal. (1999) 9: 317-326.
[93]	A general regularity theory for stable codimension 1 integral varifolds. Ann. of Math. (2) (2014) 179: 843-1007.
[94]	Stable and finite Morse index solutions of a nonlinear elliptic system. J. Math. Anal. Appl. (2019) 471: 147-169.

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