American Institute of Mathematical Sciences

December  2021, 29(6): 4215-4228. doi: 10.3934/era.2021080

On the number of critical points of solutions of semilinear elliptic equations

 Dipartimento di Matematica, Università di Roma "La Sapienza", P.le A. Moro 2 - 00185 Roma, Italy

Dedicated to Norman Dancer, a gentleman of mathematical analysis.

Received  July 2021 Revised  August 2021 Published  December 2021 Early access  October 2021

Fund Project: Partially supported by Indam-Gnampa

In this survey we discuss old and new results on the number of critical points of solutions of the problem
 $$$\begin{cases} -\Delta u = f(u)&in\ \Omega\\ u = 0&on\ \partial \Omega \end{cases} \;\;\;\;\;\;\;\;(0.1)$$$
where
 $\Omega\subset \mathbb{R}^N$
with
 $N\ge2$
is a smooth bounded domain. Both cases where
 $u$
is a positive or nodal solution will be considered.
Citation: 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
References:
 [1] A. Acker, L. E. Payne and G. Philippin, On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem, Z. Angew. Math. Phys., 32 (1981), 683-694.  doi: 10.1007/BF00946979.  Google Scholar [2] G. Alessandrini, Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains, Comment. Math. Helv., 69 (1994), 142-154.  doi: 10.1007/BF02564478.  Google Scholar [3] G. Alessandrini and R. Magnanini, The index of isolated critical points and solutions of elliptic equations in the plane, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 567-589, http://www.numdam.org/item?id=ASNSP_1992_4_19_4_567_0.  Google Scholar [4] J. Arango and A. Gómez, Critical points of solutions to elliptic problems in planar domains, Commun. Pure Appl. Anal., 10 (2011), 327-338.  doi: 10.3934/cpaa.2011.10.327.  Google Scholar [5] A. Bahri, Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations, 3 (1995), 67-93.  doi: 10.1007/BF01190892.  Google Scholar [6] P. Berard and B. Helffer, Nodal sets of eigenfunctions, Antonie Stern's results revisited, in Actes du séminaire de Théorie spectrale et géométrie, Vol. 32, Institut Fourier, Cedram, (2014-2015), 1-37. Google Scholar [7] H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis, 22 (1976), 366-389.  doi: 10.1016/0022-1236(76)90004-5.  Google Scholar [8] X. Cabré and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.), 4 (1998), 1-10.  doi: 10.1007/s000290050022.  Google Scholar [9] D. Cao, N. E. Dancer, E. S. Noussair and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete Contin. Dynam. Systems, 2 (1996), 221-236.  doi: 10.3934/dcds.1996.2.221.  Google Scholar [10] M. Clapp, M. Musso and A. Pistoia, Multipeak solutions to the Bahri-Coron problem in domains with a shrinking hole, J. Funct. Anal., 256 (2009), 275-306.  doi: 10.1016/j.jfa.2008.06.034.  Google Scholar [11] J. Dahne, J. Gómez-Serrano and K. Hou, A counterexample to payne's nodal line conjecture with few holes, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), Paper No. 105957, 13 pp. doi: 10.1016/j.cnsns.2021.105957.  Google Scholar [12] L. Damascelli, On the nodal set of the second eigenfunction of the Laplacian in symmetric domains in $\Bbb R^N$, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11 (2000), 175-181.   Google Scholar [13] E. N. Dancer and J. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes, Manuscripta Math., 123 (2007), 493-511.  doi: 10.1007/s00229-007-0110-6.  Google Scholar [14] F. De Regibus and M. Grossi, On the number of critical points of stable solutions in bounded strip-like domains, 2021. Google Scholar [15] F. De Regibus, M. Grossi and D. Mukherjee, Uniqueness of the critical point for semi-stable solutions in $\Bbb R^2$, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 25, 13 pp. doi: 10.1007/s00526-020-01903-5.  Google Scholar [16] M. del Pino, P. Felmer and M. Musso, Multi-peak solutions for super-critical elliptic problems in domains with small holes, J. Differential Equations, 182 (2002), 511-540.  doi: 10.1006/jdeq.2001.4098.  Google Scholar [17] M. del Pino and J. Wei, Problèmes elliptiques supercritiques dans des domaines avec de petits trous, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 507-520.  doi: 10.1016/j.anihpc.2006.03.001.  Google Scholar [18] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar [19] F. Gladiali and M. Grossi, On the number of critical points of solutions of semilinear equations in $\mathbb{R}^2$, to appear in Amer. Jour. Math.. Google Scholar [20] F. Gladiali and M. Grossi, Strict convexity of level sets of solutions of some nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 363-373.  doi: 10.1017/S0308210500003255.  Google Scholar [21] D. Grieser and D. Jerison, Asymptotics of the first nodal line of a convex domain, Invent. Math., 125 (1996), 197-219.  doi: 10.1007/s002220050073.  Google Scholar [22] M. Grossi and P. Luo, On the number and location of critical points of solutions of nonlinear elliptic equations in domains with a small hole, 2020. Google Scholar [23] M. Grossi and R. Molle, On the shape of the solutions of some semilinear elliptic problems, Commun. Contemp. Math., 5 (2003), 85-99.  doi: 10.1142/S0219199703000914.  Google Scholar [24] F. Hamel, N. Nadirashvili and Y. Sire, Convexity of level sets for elliptic problems in convex domains or convex rings: Two counterexamples, Amer. J. Math., 138 (2016), 499-527.  doi: 10.1353/ajm.2016.0012.  Google Scholar [25] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and N. Nadirashvili, The nodal line of the second eigenfunction of the laplacian in $\Bbb R^2$ can be closed, Duke Math. J., 90 (1997), 631-640.  doi: 10.1215/S0012-7094-97-09017-7.  Google Scholar [26] D. Jerison, The diameter of the first nodal line of a convex domain, Ann. of Math. (2), 141 (1995), 1-33.  doi: 10.2307/2118626.  Google Scholar [27] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, vol. 1150 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075060.  Google Scholar [28] G. Li, S. Yan and J. Yang, An elliptic problem with critical growth in domains with shrinking holes, J. Differential Equations, 198 (2004), 275-300.  doi: 10.1016/j.jde.2003.06.001.  Google Scholar [29] Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations, 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar [30] C. S. Lin, On the second eigenfunctions of the Laplacian in $\mathbb{R}^2$, Comm. Math. Phys., 111 (1987), 161-166. http://projecteuclid.org/euclid.cmp/1104159536. doi: 10.1007/BF01217758.  Google Scholar [31] L. G. Makar-Limanov, The solution of the Dirichlet problem for the equation $\Delta u = -1$ in a convex region, Mat. Zametki, 9 (1971), 89-92.   Google Scholar [32] A. D. Melas, On the nodal line of the second eigenfunction of the Laplacian in $\mathbb{R}^2$, J. Differential Geom., 35 (1992), 255-263, http://projecteuclid.org/euclid.jdg/1214447811.  Google Scholar [33] M. Morse and G. B. Van Schaack, The critical point theory under general boundary conditions, Ann. of Math. (2), 35 (1934), 545-571.  doi: 10.2307/1968750.  Google Scholar [34] F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities, J. Funct. Anal., 192 (2002), 271-282.  doi: 10.1006/jfan.2001.3901.  Google Scholar [35] L. E. Payne, Isoperimetric inequalities and their applications, SIAM Rev., 9 (1967), 453-488.  doi: 10.1137/1009070.  Google Scholar [36] L. E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys., 24 (1973), 721-729.  doi: 10.1007/BF01597076.  Google Scholar [37] L. Qi, Extrema of a real polynomial, J. Global Optim., 30 (2004), 405-433.  doi: 10.1007/s10898-004-6875-1.  Google Scholar [38] O. Rey, Sur un problème variationnel non compact: L'effet de petits trous dans le domaine, C. R. Acad. Sci. Paris Sér. I Math., 308 (1989), 349-352.   Google Scholar [39] E. H. Rothe, A relation between the type numbers of a critical point and the index of the corresponding field of gradient vectors, Math. Nachr., 4 (1951), 12-17.   Google Scholar [40] A. Stern, Bemerkungen über Asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, PhD Thesis, Druck der Dieterichschen UniversitätsBuchdruckerei (W. Fr. Kaestner), Göttingen, Germany, 1925. Google Scholar [41] H. Whitney, A function not constant on a connected set of critical points, Duke Math. J., 1 (1935), 514-517.  doi: 10.1215/S0012-7094-35-00138-7.  Google Scholar [42] S. T. Yau, Problem section, in Seminar on Differential Geometry, vol. 102 of Ann. of Math. Stud., Princeton Univ. Press, Princeton, N.J., 1982, 669–706.  Google Scholar

show all references

References:
 [1] A. Acker, L. E. Payne and G. Philippin, On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem, Z. Angew. Math. Phys., 32 (1981), 683-694.  doi: 10.1007/BF00946979.  Google Scholar [2] G. Alessandrini, Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains, Comment. Math. Helv., 69 (1994), 142-154.  doi: 10.1007/BF02564478.  Google Scholar [3] G. Alessandrini and R. Magnanini, The index of isolated critical points and solutions of elliptic equations in the plane, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 567-589, http://www.numdam.org/item?id=ASNSP_1992_4_19_4_567_0.  Google Scholar [4] J. Arango and A. Gómez, Critical points of solutions to elliptic problems in planar domains, Commun. Pure Appl. Anal., 10 (2011), 327-338.  doi: 10.3934/cpaa.2011.10.327.  Google Scholar [5] A. Bahri, Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations, 3 (1995), 67-93.  doi: 10.1007/BF01190892.  Google Scholar [6] P. Berard and B. Helffer, Nodal sets of eigenfunctions, Antonie Stern's results revisited, in Actes du séminaire de Théorie spectrale et géométrie, Vol. 32, Institut Fourier, Cedram, (2014-2015), 1-37. Google Scholar [7] H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis, 22 (1976), 366-389.  doi: 10.1016/0022-1236(76)90004-5.  Google Scholar [8] X. Cabré and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.), 4 (1998), 1-10.  doi: 10.1007/s000290050022.  Google Scholar [9] D. Cao, N. E. Dancer, E. S. Noussair and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete Contin. Dynam. Systems, 2 (1996), 221-236.  doi: 10.3934/dcds.1996.2.221.  Google Scholar [10] M. Clapp, M. Musso and A. Pistoia, Multipeak solutions to the Bahri-Coron problem in domains with a shrinking hole, J. Funct. Anal., 256 (2009), 275-306.  doi: 10.1016/j.jfa.2008.06.034.  Google Scholar [11] J. Dahne, J. Gómez-Serrano and K. Hou, A counterexample to payne's nodal line conjecture with few holes, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), Paper No. 105957, 13 pp. doi: 10.1016/j.cnsns.2021.105957.  Google Scholar [12] L. Damascelli, On the nodal set of the second eigenfunction of the Laplacian in symmetric domains in $\Bbb R^N$, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11 (2000), 175-181.   Google Scholar [13] E. N. Dancer and J. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes, Manuscripta Math., 123 (2007), 493-511.  doi: 10.1007/s00229-007-0110-6.  Google Scholar [14] F. De Regibus and M. Grossi, On the number of critical points of stable solutions in bounded strip-like domains, 2021. Google Scholar [15] F. De Regibus, M. Grossi and D. Mukherjee, Uniqueness of the critical point for semi-stable solutions in $\Bbb R^2$, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 25, 13 pp. doi: 10.1007/s00526-020-01903-5.  Google Scholar [16] M. del Pino, P. Felmer and M. Musso, Multi-peak solutions for super-critical elliptic problems in domains with small holes, J. Differential Equations, 182 (2002), 511-540.  doi: 10.1006/jdeq.2001.4098.  Google Scholar [17] M. del Pino and J. Wei, Problèmes elliptiques supercritiques dans des domaines avec de petits trous, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 507-520.  doi: 10.1016/j.anihpc.2006.03.001.  Google Scholar [18] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar [19] F. Gladiali and M. Grossi, On the number of critical points of solutions of semilinear equations in $\mathbb{R}^2$, to appear in Amer. Jour. Math.. Google Scholar [20] F. Gladiali and M. Grossi, Strict convexity of level sets of solutions of some nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 363-373.  doi: 10.1017/S0308210500003255.  Google Scholar [21] D. Grieser and D. Jerison, Asymptotics of the first nodal line of a convex domain, Invent. Math., 125 (1996), 197-219.  doi: 10.1007/s002220050073.  Google Scholar [22] M. Grossi and P. Luo, On the number and location of critical points of solutions of nonlinear elliptic equations in domains with a small hole, 2020. Google Scholar [23] M. Grossi and R. Molle, On the shape of the solutions of some semilinear elliptic problems, Commun. Contemp. Math., 5 (2003), 85-99.  doi: 10.1142/S0219199703000914.  Google Scholar [24] F. Hamel, N. Nadirashvili and Y. Sire, Convexity of level sets for elliptic problems in convex domains or convex rings: Two counterexamples, Amer. J. Math., 138 (2016), 499-527.  doi: 10.1353/ajm.2016.0012.  Google Scholar [25] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and N. Nadirashvili, The nodal line of the second eigenfunction of the laplacian in $\Bbb R^2$ can be closed, Duke Math. J., 90 (1997), 631-640.  doi: 10.1215/S0012-7094-97-09017-7.  Google Scholar [26] D. Jerison, The diameter of the first nodal line of a convex domain, Ann. of Math. (2), 141 (1995), 1-33.  doi: 10.2307/2118626.  Google Scholar [27] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, vol. 1150 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075060.  Google Scholar [28] G. Li, S. Yan and J. Yang, An elliptic problem with critical growth in domains with shrinking holes, J. Differential Equations, 198 (2004), 275-300.  doi: 10.1016/j.jde.2003.06.001.  Google Scholar [29] Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations, 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar [30] C. S. Lin, On the second eigenfunctions of the Laplacian in $\mathbb{R}^2$, Comm. Math. Phys., 111 (1987), 161-166. http://projecteuclid.org/euclid.cmp/1104159536. doi: 10.1007/BF01217758.  Google Scholar [31] L. G. Makar-Limanov, The solution of the Dirichlet problem for the equation $\Delta u = -1$ in a convex region, Mat. Zametki, 9 (1971), 89-92.   Google Scholar [32] A. D. Melas, On the nodal line of the second eigenfunction of the Laplacian in $\mathbb{R}^2$, J. Differential Geom., 35 (1992), 255-263, http://projecteuclid.org/euclid.jdg/1214447811.  Google Scholar [33] M. Morse and G. B. Van Schaack, The critical point theory under general boundary conditions, Ann. of Math. (2), 35 (1934), 545-571.  doi: 10.2307/1968750.  Google Scholar [34] F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities, J. Funct. Anal., 192 (2002), 271-282.  doi: 10.1006/jfan.2001.3901.  Google Scholar [35] L. E. Payne, Isoperimetric inequalities and their applications, SIAM Rev., 9 (1967), 453-488.  doi: 10.1137/1009070.  Google Scholar [36] L. E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys., 24 (1973), 721-729.  doi: 10.1007/BF01597076.  Google Scholar [37] L. Qi, Extrema of a real polynomial, J. Global Optim., 30 (2004), 405-433.  doi: 10.1007/s10898-004-6875-1.  Google Scholar [38] O. Rey, Sur un problème variationnel non compact: L'effet de petits trous dans le domaine, C. R. Acad. Sci. Paris Sér. I Math., 308 (1989), 349-352.   Google Scholar [39] E. H. Rothe, A relation between the type numbers of a critical point and the index of the corresponding field of gradient vectors, Math. Nachr., 4 (1951), 12-17.   Google Scholar [40] A. Stern, Bemerkungen über Asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, PhD Thesis, Druck der Dieterichschen UniversitätsBuchdruckerei (W. Fr. Kaestner), Göttingen, Germany, 1925. Google Scholar [41] H. Whitney, A function not constant on a connected set of critical points, Duke Math. J., 1 (1935), 514-517.  doi: 10.1215/S0012-7094-35-00138-7.  Google Scholar [42] S. T. Yau, Problem section, in Seminar on Differential Geometry, vol. 102 of Ann. of Math. Stud., Princeton Univ. Press, Princeton, N.J., 1982, 669–706.  Google Scholar
A picture of D with $c = \frac1{300000}$
The domain $\Omega$ in Theorem 3.2
Domain $\Omega_ \varepsilon$ with $k = 2$ and level set $u_ \varepsilon = c$
Nodal line of an eigenfuncion in the rectangle (Stern's PhD thesis)
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