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

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

Dedicated to Professor Norman Dancer on the occasion of his 75th birthday

Received  June 2021 Early access August 2021

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.

Citation: Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, doi: 10.3934/era.2021062
References:
[1]

O. AgudeloM. del Pino and J. Wei, Higher-dimensional catenoid, Liouville equation, and Allen-Cahn equation, Int. Math. Res. Not. IMRN, 2016 (2016), 7051-7102.  doi: 10.1093/imrn/rnv350.  Google Scholar

[2]

H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar

[3]

H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math., 368 (1986), 63-107.   Google Scholar

[4]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbb{R}^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3.  Google Scholar

[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).  Google Scholar

[6]

X. CabréE. Cinti and J. Serra, Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s\sim 1$, J. Reine Angew. Math., 764 (2020), 157-180.  doi: 10.1515/crelle-2019-0005.  Google Scholar

[7]

X. CabréA. FigalliX. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math., 224 (2020), 187-252.  doi: 10.4310/ACTA.2020.v224.n2.a1.  Google Scholar

[8]

L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

[9]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871.  doi: 10.1016/j.aim.2013.08.007.  Google Scholar

[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.  Google Scholar

[11]

H.-D. CaoY. Shen and S. Zhu, The structure of stable minimal hypersurfaces in $\mathbb{R}^{n+1}$, Math. Res. Lett., 4 (1997), 637-644.  doi: 10.4310/MRL.1997.v4.n5.a2.  Google Scholar

[12]

O. Chodosh and C. Mantoulidis, Minimal surfaces and the Allen-Cahn equation on 3-manifolds: Index, multiplicity, and curvature estimates, Ann. of Math. (2), 191 (2020), 213-328.  doi: 10.4007/annals.2020.191.1.4.  Google Scholar

[13]

E. N. Dancer, Stable and finite Morse index solutions on $\mathbb{R}^n$ or on bounded domains with small diffusion. II, Indiana Univ. Math. J., 53 (2004), 97-108.  doi: 10.1512/iumj.2004.53.2354.  Google Scholar

[14]

E. N. Dancer, Stable solutions on $\mathbb{R}^n$ and the primary branch of some non-self-adjoint convex problems, Differential Integral Equations, 17 (2004), 961-970.   Google Scholar

[15]

E. N. Dancer, Stable and finite Morse index solutions on $\mathbb{R}^n$ or on bounded domains with small diffusion, Trans. Amer. Math. Soc., 357 (2005), 1225-1243.  doi: 10.1090/S0002-9947-04-03543-3.  Google Scholar

[16]

E. N. Dancer, Supercritical finite Morse index solutions, Nonlinear Anal., 66 (2007), 268-269.  doi: 10.1016/j.na.2005.11.012.  Google Scholar

[17]

E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 173-179.  doi: 10.1016/j.anihpc.2006.12.001.  Google Scholar

[18]

E. N. Dancer, Finite Morse index solutions of supercritical problems, J. Reine Angew. Math., 620 (2008), 213-233.  doi: 10.1515/CRELLE.2008.055.  Google Scholar

[19]

E. N. Dancer, 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., 9 (2009), 657-678.  doi: 10.1515/ans-2009-0405.  Google Scholar

[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).  Google Scholar

[21]

E. N. Dancer, New results for finite Morse index solutions on $\mathbb{R}^N$ and applications, Adv. Nonlinear Stud., 10 (2010), 581-595.  doi: 10.1515/ans-2010-0304.  Google Scholar

[22]

E. N. DancerY. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005.  Google Scholar

[23]

E. N. Dancer and A. Farina, 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., 137 (2009), 1333-1338.  doi: 10.1090/S0002-9939-08-09772-4.  Google Scholar

[24]

E. N. DancerZ. Guo and J. Wei, Non-radial singular solutions of the Lane-Emden equation in $\mathbb{R}^N$, Indiana Univ. Math. J., 61 (2012), 1971-1996.  doi: 10.1512/iumj.2012.61.4749.  Google Scholar

[25]

J. DávilaL. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232.  doi: 10.1016/j.jfa.2010.12.028.  Google Scholar

[26]

J. DávilaL. DupaigneK. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.  Google Scholar

[27]

J. DávilaL. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087-6104.  doi: 10.1090/tran/6872.  Google Scholar

[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).  Google Scholar

[29]

D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math., 635 (2009), 1-21.  doi: 10.1515/CRELLE.2009.074.  Google Scholar

[30]

M. del PinoM. KowalczykF. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$, J. Funct. Anal., 258 (2010), 458-503.  doi: 10.1016/j.jfa.2009.04.020.  Google Scholar

[31]

M. del PinoM. Kowalczyk and J. Wei, The Toda system and clustering interfaces in the Allen-Cahn equation, Arch. Ration. Mech. Anal., 190 (2008), 141-187.  doi: 10.1007/s00205-008-0143-3.  Google Scholar

[32]

M. del PinoM. Kowalczyk and J. Wei, On De Giorgi's conjecture in dimension $N\geq 9$, Ann. of Math. (2), 174 (2011), 1485-1569.  doi: 10.4007/annals.2011.174.3.3.  Google Scholar

[33]

M. del PinoM. Kowalczyk and J. Wei, Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$, J. Differential Geom., 93 (2013), 67-131.   Google Scholar

[34]

M. del PinoM. KowalczykJ. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature, Geom. Funct. Anal., 20 (2010), 918-957.  doi: 10.1007/s00039-010-0083-6.  Google Scholar

[35]

Y. Du, The work of Norman Dancer, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 1807-1833.  doi: 10.3934/dcdss.2019119.  Google Scholar

[36]

Y. Du and Z. Guo, Positive solutions of an elliptic equation with negative exponent: Stability and critical power, J. Differential Equations, 246 (2009), 2387-2414.  doi: 10.1016/j.jde.2008.08.008.  Google Scholar

[37]

Y. Du and Z. Guo, Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic equations, Adv. Differential Equations, 18 (2013), 737-768.   Google Scholar

[38]

Y. Du and Z. Guo, Finite Morse index solutions of weighted elliptic equations and the critical exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3161-3181.  doi: 10.1007/s00526-015-0897-z.  Google Scholar

[39]

Y. DuZ. Guo and K. Wang, Monotonicity formula and $\varepsilon$-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions, Calc. Var. Partial Differential Equations, 50 (2014), 615-638.  doi: 10.1007/s00526-013-0649-x.  Google Scholar

[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.  Google Scholar

[41]

L. Dupaigne and A. Farina, Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains, arXiv: Analysis of PDEs, (2019). Google Scholar

[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.  Google Scholar

[43]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[44]

A. Farina, Stable solutions of $-\Delta u = e^u$ on $\mathbb{R}^N$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

M. Fazly and J. Wei, On finite Morse index solutions of higher order fractional Lane-Emden equations, Amer. J. Math., 139 (2017), 433-460.  doi: 10.1353/ajm.2017.0011.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[50]

H. Federer, 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., 76 (1970), 767-771.  doi: 10.1090/S0002-9904-1970-12542-3.  Google Scholar

[51]

X. Fernández-Real and X. Ros-Oton, On global solutions to semilinear elliptic equations related to the one-phase free boundary problem, Discrete Contin. Dyn. Syst., 39 (2019), 6945-6959.  doi: 10.3934/dcds.2019238.  Google Scholar

[52]

A. Figalli and J. Serra, On stable solutions for boundary reactions: A De Giorgi-type result in dimension $4+1$, Invent. Math., 219 (2020), 153-177.  doi: 10.1007/s00222-019-00904-2.  Google Scholar

[53]

W. H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2), 11 (1962), 69-90.  doi: 10.1007/BF02849427.  Google Scholar

[54]

F. Gazzola, The sharp exponent for a Liouville-type theorem for an elliptic inequality, Rend. Istit. Mat. Univ. Trieste, 34 (2003), 99-102.   Google Scholar

[55]

N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491.  doi: 10.1007/s002080050196.  Google Scholar

[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.  Google Scholar

[57]

C. GuiK. Wang and J. Wei, Axially symmetric solutions of the Allen-Cahn equation with finite Morse index, Trans. Amer. Math. Soc., 373 (2020), 3649-3668.  doi: 10.1090/tran/8035.  Google Scholar

[58]

Z. Guo, On the symmetry of positive solutions of the Lane-Emden equation with supercritical exponent, Adv. Differential Equations, 7 (2002), 641-666.   Google Scholar

[59]

Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580.  doi: 10.3934/dcds.2014.34.2561.  Google Scholar

[60]

L.-G. Hu, Liouville-type theorems for the fourth order nonlinear elliptic equation, J. Differential Equations, 256 (2014), 1817-1846.  doi: 10.1016/j.jde.2013.12.001.  Google Scholar

[61]

L.-G. Hu, A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system, Adv. Differential Equations, 22 (2017), 49-76.   Google Scholar

[62]

X. HuangD. Ye and F. Zhou, Stability for entire radial solutions to the biharmonic equation with negative exponents, C. R. Math. Acad. Sci. Paris, 356 (2018), 632-636.  doi: 10.1016/j.crma.2018.05.001.  Google Scholar

[63]

A. Hyder and W. Yang, Classification of stable solutions to a non-local Gelfand-Liouville equation, arXiv: Analysis of PDEs, (2020). Google Scholar

[64]

D. Jerison and K. Perera, Higher critical points in an elliptic free boundary problem, J. Geom. Anal., 28 (2018), 1258-1294.  doi: 10.1007/s12220-017-9862-8.  Google Scholar

[65]

D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal., 25 (2015), 1240-1257.  doi: 10.1007/s00039-015-0335-6.  Google Scholar

[66]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.  doi: 10.1007/BF00250508.  Google Scholar

[67]

B. Lai, The regularity and stability of solutions to semilinear fourth-order elliptic problems with negative exponents, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 195-212.  doi: 10.1017/S0308210515000426.  Google Scholar

[68]

B. Lai and D. Ye, Remarks on entire solutions for two fourth-order elliptic problems, Proc. Edinb. Math. Soc. (2), 59 (2016), 777-786.  doi: 10.1017/S0013091515000371.  Google Scholar

[69]

P. Li and J. Wang, Minimal hypersurfaces with finite index, Math. Res. Lett., 9 (2002), 95-103.  doi: 10.4310/MRL.2002.v9.n1.a7.  Google Scholar

[70]

Senping Luo, Juncheng Wei and Wenming Zou, Monotonicity formula and classification of stable solutions to polyharmonic Lane-Emden equations, 2020. Google Scholar

[71]

L. Ma and J. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., 254 (2008), 1058-1087.  doi: 10.1016/j.jfa.2007.09.017.  Google Scholar

[72]

A. M. Meadows, Stable and singular solutions of the equation $\Delta u = 1/u$, Indiana Univ. Math. J., 53 (2004), 1681-1703.  doi: 10.1512/iumj.2004.53.2560.  Google Scholar

[73]

F. Mtiri and D. Ye, Liouville theorems for stable at infinity solutions of Lane-Emden system, Nonlinearity, 32 (2019), 910-926.  doi: 10.1088/1361-6544/aaf078.  Google Scholar

[74]

F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math., 79 (1993), 161-172.  doi: 10.1007/BF02568335.  Google Scholar

[75]

F. Pacard, Convergence and partial regularity for weak solutions of some nonlinear elliptic equation: The supercritical case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 537-551.  doi: 10.1016/S0294-1449(16)30177-9.  Google Scholar

[76]

F. Pacard and J. Wei, Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, J. Funct. Anal., 264 (2013), 1131-1167.  doi: 10.1016/j.jfa.2012.03.010.  Google Scholar

[77]

D. Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J., 32 (1983), 1-17.  doi: 10.1512/iumj.1983.32.32001.  Google Scholar

[78]

O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2), 169 (2009), 41-78.  doi: 10.4007/annals.2009.169.41.  Google Scholar

[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).  Google Scholar

[80]

R. Schoen and L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math., 34 (1981), 741-797.  doi: 10.1002/cpa.3160340603.  Google Scholar

[81]

R. SchoenL. Simon and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math., 134 (1975), 275-288.  doi: 10.1007/BF02392104.  Google Scholar

[82]

J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2), 88 (1968), 62-105.  doi: 10.2307/1970556.  Google Scholar

[83]

F. Takahashi, Topics of stable solutions to elliptic equations, Sugaku Expositions, 34 (2021), 35-59.  doi: 10.1090/suga/457.  Google Scholar

[84]

C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal., 262 (2012), 1705-1727.  doi: 10.1016/j.jfa.2011.11.017.  Google Scholar

[85]

K. Wang, Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 44 (2012), 601-610.  doi: 10.1007/s00526-011-0446-3.  Google Scholar

[86]

K. Wang, Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260.  doi: 10.1016/j.na.2012.04.041.  Google Scholar

[87]

K. Wang, Erratum to: Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 47 (2013), 433-435.  doi: 10.1007/s00526-012-0565-5.  Google Scholar

[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. Google Scholar

[89]

K. Wang, Stable and finite Morse index solutions of Toda system, J. Differential Equations, 268 (2019), 60-79.  doi: 10.1016/j.jde.2019.08.006.  Google Scholar

[90]

K. Wang and J. Wei, Finite Morse index implies finite ends, Comm. Pure Appl. Math., 72 (2019), 1044-1119.  doi: 10.1002/cpa.21812.  Google Scholar

[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.  Google Scholar

[92]

G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal., 9 (1999), 317-326.  doi: 10.1007/BF02921941.  Google Scholar

[93]

N. Wickramasekera, A general regularity theory for stable codimension 1 integral varifolds, Ann. of Math. (2), 179 (2014), 843-1007.  doi: 10.4007/annals.2014.179.3.2.  Google Scholar

[94]

H. Yang and W. Zou, Stable and finite Morse index solutions of a nonlinear elliptic system, J. Math. Anal. Appl., 471 (2019), 147-169.  doi: 10.1016/j.jmaa.2018.10.069.  Google Scholar

show all references

References:
[1]

O. AgudeloM. del Pino and J. Wei, Higher-dimensional catenoid, Liouville equation, and Allen-Cahn equation, Int. Math. Res. Not. IMRN, 2016 (2016), 7051-7102.  doi: 10.1093/imrn/rnv350.  Google Scholar

[2]

H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar

[3]

H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math., 368 (1986), 63-107.   Google Scholar

[4]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbb{R}^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3.  Google Scholar

[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).  Google Scholar

[6]

X. CabréE. Cinti and J. Serra, Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s\sim 1$, J. Reine Angew. Math., 764 (2020), 157-180.  doi: 10.1515/crelle-2019-0005.  Google Scholar

[7]

X. CabréA. FigalliX. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math., 224 (2020), 187-252.  doi: 10.4310/ACTA.2020.v224.n2.a1.  Google Scholar

[8]

L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

[9]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871.  doi: 10.1016/j.aim.2013.08.007.  Google Scholar

[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.  Google Scholar

[11]

H.-D. CaoY. Shen and S. Zhu, The structure of stable minimal hypersurfaces in $\mathbb{R}^{n+1}$, Math. Res. Lett., 4 (1997), 637-644.  doi: 10.4310/MRL.1997.v4.n5.a2.  Google Scholar

[12]

O. Chodosh and C. Mantoulidis, Minimal surfaces and the Allen-Cahn equation on 3-manifolds: Index, multiplicity, and curvature estimates, Ann. of Math. (2), 191 (2020), 213-328.  doi: 10.4007/annals.2020.191.1.4.  Google Scholar

[13]

E. N. Dancer, Stable and finite Morse index solutions on $\mathbb{R}^n$ or on bounded domains with small diffusion. II, Indiana Univ. Math. J., 53 (2004), 97-108.  doi: 10.1512/iumj.2004.53.2354.  Google Scholar

[14]

E. N. Dancer, Stable solutions on $\mathbb{R}^n$ and the primary branch of some non-self-adjoint convex problems, Differential Integral Equations, 17 (2004), 961-970.   Google Scholar

[15]

E. N. Dancer, Stable and finite Morse index solutions on $\mathbb{R}^n$ or on bounded domains with small diffusion, Trans. Amer. Math. Soc., 357 (2005), 1225-1243.  doi: 10.1090/S0002-9947-04-03543-3.  Google Scholar

[16]

E. N. Dancer, Supercritical finite Morse index solutions, Nonlinear Anal., 66 (2007), 268-269.  doi: 10.1016/j.na.2005.11.012.  Google Scholar

[17]

E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 173-179.  doi: 10.1016/j.anihpc.2006.12.001.  Google Scholar

[18]

E. N. Dancer, Finite Morse index solutions of supercritical problems, J. Reine Angew. Math., 620 (2008), 213-233.  doi: 10.1515/CRELLE.2008.055.  Google Scholar

[19]

E. N. Dancer, 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., 9 (2009), 657-678.  doi: 10.1515/ans-2009-0405.  Google Scholar

[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).  Google Scholar

[21]

E. N. Dancer, New results for finite Morse index solutions on $\mathbb{R}^N$ and applications, Adv. Nonlinear Stud., 10 (2010), 581-595.  doi: 10.1515/ans-2010-0304.  Google Scholar

[22]

E. N. DancerY. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005.  Google Scholar

[23]

E. N. Dancer and A. Farina, 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., 137 (2009), 1333-1338.  doi: 10.1090/S0002-9939-08-09772-4.  Google Scholar

[24]

E. N. DancerZ. Guo and J. Wei, Non-radial singular solutions of the Lane-Emden equation in $\mathbb{R}^N$, Indiana Univ. Math. J., 61 (2012), 1971-1996.  doi: 10.1512/iumj.2012.61.4749.  Google Scholar

[25]

J. DávilaL. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232.  doi: 10.1016/j.jfa.2010.12.028.  Google Scholar

[26]

J. DávilaL. DupaigneK. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.  Google Scholar

[27]

J. DávilaL. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087-6104.  doi: 10.1090/tran/6872.  Google Scholar

[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).  Google Scholar

[29]

D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math., 635 (2009), 1-21.  doi: 10.1515/CRELLE.2009.074.  Google Scholar

[30]

M. del PinoM. KowalczykF. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$, J. Funct. Anal., 258 (2010), 458-503.  doi: 10.1016/j.jfa.2009.04.020.  Google Scholar

[31]

M. del PinoM. Kowalczyk and J. Wei, The Toda system and clustering interfaces in the Allen-Cahn equation, Arch. Ration. Mech. Anal., 190 (2008), 141-187.  doi: 10.1007/s00205-008-0143-3.  Google Scholar

[32]

M. del PinoM. Kowalczyk and J. Wei, On De Giorgi's conjecture in dimension $N\geq 9$, Ann. of Math. (2), 174 (2011), 1485-1569.  doi: 10.4007/annals.2011.174.3.3.  Google Scholar

[33]

M. del PinoM. Kowalczyk and J. Wei, Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$, J. Differential Geom., 93 (2013), 67-131.   Google Scholar

[34]

M. del PinoM. KowalczykJ. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature, Geom. Funct. Anal., 20 (2010), 918-957.  doi: 10.1007/s00039-010-0083-6.  Google Scholar

[35]

Y. Du, The work of Norman Dancer, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 1807-1833.  doi: 10.3934/dcdss.2019119.  Google Scholar

[36]

Y. Du and Z. Guo, Positive solutions of an elliptic equation with negative exponent: Stability and critical power, J. Differential Equations, 246 (2009), 2387-2414.  doi: 10.1016/j.jde.2008.08.008.  Google Scholar

[37]

Y. Du and Z. Guo, Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic equations, Adv. Differential Equations, 18 (2013), 737-768.   Google Scholar

[38]

Y. Du and Z. Guo, Finite Morse index solutions of weighted elliptic equations and the critical exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3161-3181.  doi: 10.1007/s00526-015-0897-z.  Google Scholar

[39]

Y. DuZ. Guo and K. Wang, Monotonicity formula and $\varepsilon$-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions, Calc. Var. Partial Differential Equations, 50 (2014), 615-638.  doi: 10.1007/s00526-013-0649-x.  Google Scholar

[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.  Google Scholar

[41]

L. Dupaigne and A. Farina, Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains, arXiv: Analysis of PDEs, (2019). Google Scholar

[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.  Google Scholar

[43]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[44]

A. Farina, Stable solutions of $-\Delta u = e^u$ on $\mathbb{R}^N$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

M. Fazly and J. Wei, On finite Morse index solutions of higher order fractional Lane-Emden equations, Amer. J. Math., 139 (2017), 433-460.  doi: 10.1353/ajm.2017.0011.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[50]

H. Federer, 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., 76 (1970), 767-771.  doi: 10.1090/S0002-9904-1970-12542-3.  Google Scholar

[51]

X. Fernández-Real and X. Ros-Oton, On global solutions to semilinear elliptic equations related to the one-phase free boundary problem, Discrete Contin. Dyn. Syst., 39 (2019), 6945-6959.  doi: 10.3934/dcds.2019238.  Google Scholar

[52]

A. Figalli and J. Serra, On stable solutions for boundary reactions: A De Giorgi-type result in dimension $4+1$, Invent. Math., 219 (2020), 153-177.  doi: 10.1007/s00222-019-00904-2.  Google Scholar

[53]

W. H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2), 11 (1962), 69-90.  doi: 10.1007/BF02849427.  Google Scholar

[54]

F. Gazzola, The sharp exponent for a Liouville-type theorem for an elliptic inequality, Rend. Istit. Mat. Univ. Trieste, 34 (2003), 99-102.   Google Scholar

[55]

N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491.  doi: 10.1007/s002080050196.  Google Scholar

[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.  Google Scholar

[57]

C. GuiK. Wang and J. Wei, Axially symmetric solutions of the Allen-Cahn equation with finite Morse index, Trans. Amer. Math. Soc., 373 (2020), 3649-3668.  doi: 10.1090/tran/8035.  Google Scholar

[58]

Z. Guo, On the symmetry of positive solutions of the Lane-Emden equation with supercritical exponent, Adv. Differential Equations, 7 (2002), 641-666.   Google Scholar

[59]

Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580.  doi: 10.3934/dcds.2014.34.2561.  Google Scholar

[60]

L.-G. Hu, Liouville-type theorems for the fourth order nonlinear elliptic equation, J. Differential Equations, 256 (2014), 1817-1846.  doi: 10.1016/j.jde.2013.12.001.  Google Scholar

[61]

L.-G. Hu, A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system, Adv. Differential Equations, 22 (2017), 49-76.   Google Scholar

[62]

X. HuangD. Ye and F. Zhou, Stability for entire radial solutions to the biharmonic equation with negative exponents, C. R. Math. Acad. Sci. Paris, 356 (2018), 632-636.  doi: 10.1016/j.crma.2018.05.001.  Google Scholar

[63]

A. Hyder and W. Yang, Classification of stable solutions to a non-local Gelfand-Liouville equation, arXiv: Analysis of PDEs, (2020). Google Scholar

[64]

D. Jerison and K. Perera, Higher critical points in an elliptic free boundary problem, J. Geom. Anal., 28 (2018), 1258-1294.  doi: 10.1007/s12220-017-9862-8.  Google Scholar

[65]

D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal., 25 (2015), 1240-1257.  doi: 10.1007/s00039-015-0335-6.  Google Scholar

[66]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.  doi: 10.1007/BF00250508.  Google Scholar

[67]

B. Lai, The regularity and stability of solutions to semilinear fourth-order elliptic problems with negative exponents, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 195-212.  doi: 10.1017/S0308210515000426.  Google Scholar

[68]

B. Lai and D. Ye, Remarks on entire solutions for two fourth-order elliptic problems, Proc. Edinb. Math. Soc. (2), 59 (2016), 777-786.  doi: 10.1017/S0013091515000371.  Google Scholar

[69]

P. Li and J. Wang, Minimal hypersurfaces with finite index, Math. Res. Lett., 9 (2002), 95-103.  doi: 10.4310/MRL.2002.v9.n1.a7.  Google Scholar

[70]

Senping Luo, Juncheng Wei and Wenming Zou, Monotonicity formula and classification of stable solutions to polyharmonic Lane-Emden equations, 2020. Google Scholar

[71]

L. Ma and J. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., 254 (2008), 1058-1087.  doi: 10.1016/j.jfa.2007.09.017.  Google Scholar

[72]

A. M. Meadows, Stable and singular solutions of the equation $\Delta u = 1/u$, Indiana Univ. Math. J., 53 (2004), 1681-1703.  doi: 10.1512/iumj.2004.53.2560.  Google Scholar

[73]

F. Mtiri and D. Ye, Liouville theorems for stable at infinity solutions of Lane-Emden system, Nonlinearity, 32 (2019), 910-926.  doi: 10.1088/1361-6544/aaf078.  Google Scholar

[74]

F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math., 79 (1993), 161-172.  doi: 10.1007/BF02568335.  Google Scholar

[75]

F. Pacard, Convergence and partial regularity for weak solutions of some nonlinear elliptic equation: The supercritical case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 537-551.  doi: 10.1016/S0294-1449(16)30177-9.  Google Scholar

[76]

F. Pacard and J. Wei, Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, J. Funct. Anal., 264 (2013), 1131-1167.  doi: 10.1016/j.jfa.2012.03.010.  Google Scholar

[77]

D. Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J., 32 (1983), 1-17.  doi: 10.1512/iumj.1983.32.32001.  Google Scholar

[78]

O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2), 169 (2009), 41-78.  doi: 10.4007/annals.2009.169.41.  Google Scholar

[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).  Google Scholar

[80]

R. Schoen and L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math., 34 (1981), 741-797.  doi: 10.1002/cpa.3160340603.  Google Scholar

[81]

R. SchoenL. Simon and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math., 134 (1975), 275-288.  doi: 10.1007/BF02392104.  Google Scholar

[82]

J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2), 88 (1968), 62-105.  doi: 10.2307/1970556.  Google Scholar

[83]

F. Takahashi, Topics of stable solutions to elliptic equations, Sugaku Expositions, 34 (2021), 35-59.  doi: 10.1090/suga/457.  Google Scholar

[84]

C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal., 262 (2012), 1705-1727.  doi: 10.1016/j.jfa.2011.11.017.  Google Scholar

[85]

K. Wang, Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 44 (2012), 601-610.  doi: 10.1007/s00526-011-0446-3.  Google Scholar

[86]

K. Wang, Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260.  doi: 10.1016/j.na.2012.04.041.  Google Scholar

[87]

K. Wang, Erratum to: Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 47 (2013), 433-435.  doi: 10.1007/s00526-012-0565-5.  Google Scholar

[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. Google Scholar

[89]

K. Wang, Stable and finite Morse index solutions of Toda system, J. Differential Equations, 268 (2019), 60-79.  doi: 10.1016/j.jde.2019.08.006.  Google Scholar

[90]

K. Wang and J. Wei, Finite Morse index implies finite ends, Comm. Pure Appl. Math., 72 (2019), 1044-1119.  doi: 10.1002/cpa.21812.  Google Scholar

[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.  Google Scholar

[92]

G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal., 9 (1999), 317-326.  doi: 10.1007/BF02921941.  Google Scholar

[93]

N. Wickramasekera, A general regularity theory for stable codimension 1 integral varifolds, Ann. of Math. (2), 179 (2014), 843-1007.  doi: 10.4007/annals.2014.179.3.2.  Google Scholar

[94]

H. Yang and W. Zou, Stable and finite Morse index solutions of a nonlinear elliptic system, J. Math. Anal. Appl., 471 (2019), 147-169.  doi: 10.1016/j.jmaa.2018.10.069.  Google Scholar

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