### Electronic Research Archive

2021, Issue 6: 4215-4228. doi: 10.3934/era.2021080
Special Issues

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

• Received: 01 July 2021 Revised: 01 August 2021 Published: 08 October 2021
• 35B05, 35B06, 35B09

• 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[J]. Electronic Research Archive, 2021, 29(6): 4215-4228. doi: 10.3934/era.2021080

### Related Papers:

• 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.

 [1] 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. (1981) 32: 683-694. [2] Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains. Comment. Math. Helv. (1994) 69: 142-154. [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. [4] Critical points of solutions to elliptic problems in planar domains. Commun. Pure Appl. Anal. (2011) 10: 327-338. [5] On a variational problem with lack of compactness: The topological effect of the critical points at infinity. Calc. Var. Partial Differential Equations (1995) 3: 67-93. [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. [7] 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 (1976) 22: 366-389. [8] Stable solutions of semilinear elliptic problems in convex domains. Selecta Math. (N.S.) (1998) 4: 1-10. [9] On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discrete Contin. Dynam. Systems (1996) 2: 221-236. [10] Multipeak solutions to the Bahri-Coron problem in domains with a shrinking hole. J. Funct. Anal. (2009) 256: 275-306. [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 [12] 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. (2000) 11: 175-181. [13] Sign-changing solutions for supercritical elliptic problems in domains with small holes. Manuscripta Math. (2007) 123: 493-511. [14] F. De Regibus and M. Grossi, On the number of critical points of stable solutions in bounded strip-like domains, 2021. [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 [16] Multi-peak solutions for super-critical elliptic problems in domains with small holes. J. Differential Equations (2002) 182: 511-540. [17] Problèmes elliptiques supercritiques dans des domaines avec de petits trous. Ann. Inst. H. Poincaré Anal. Non Linéaire (2007) 24: 507-520. [18] Symmetry and related properties via the maximum principle. Comm. Math. Phys. (1979) 68: 209-243. [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.. [20] Strict convexity of level sets of solutions of some nonlinear elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A (2004) 134: 363-373. [21] Asymptotics of the first nodal line of a convex domain. Invent. Math. (1996) 125: 197-219. [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. [23] On the shape of the solutions of some semilinear elliptic problems. Commun. Contemp. Math. (2003) 5: 85-99. [24] Convexity of level sets for elliptic problems in convex domains or convex rings: Two counterexamples. Amer. J. Math. (2016) 138: 499-527. [25] The nodal line of the second eigenfunction of the laplacian in $\Bbb R^2$ can be closed. Duke Math. J. (1997) 90: 631-640. [26] The diameter of the first nodal line of a convex domain. Ann. of Math. (2) (1995) 141: 1-33. [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 [28] An elliptic problem with critical growth in domains with shrinking holes. J. Differential Equations (2004) 198: 275-300. [29] Existence of many positive solutions of semilinear elliptic equations on annulus. J. Differential Equations (1990) 83: 348-367. [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 [31] The solution of the Dirichlet problem for the equation $\Delta u = -1$ in a convex region. Mat. Zametki (1971) 9: 89-92. [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. [33] The critical point theory under general boundary conditions. Ann. of Math. (2) (1934) 35: 545-571. [34] Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. J. Funct. Anal. (2002) 192: 271-282. [35] Isoperimetric inequalities and their applications. SIAM Rev. (1967) 9: 453-488. [36] On two conjectures in the fixed membrane eigenvalue problem. Z. Angew. Math. Phys. (1973) 24: 721-729. [37] Extrema of a real polynomial. J. Global Optim. (2004) 30: 405-433. [38] Sur un problème variationnel non compact: L'effet de petits trous dans le domaine. C. R. Acad. Sci. Paris Sér. I Math. (1989) 308: 349-352. [39] A relation between the type numbers of a critical point and the index of the corresponding field of gradient vectors. Math. Nachr. (1951) 4: 12-17. [40] A. Stern, Bemerkungen über Asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, PhD Thesis, Druck der Dieterichschen UniversitätsBuchdruckerei (W. Fr. Kaestner), Göttingen, Germany, 1925. [41] A function not constant on a connected set of critical points. Duke Math. J. (1935) 1: 514-517. [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.
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