Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in ${ℝ}^N$

Sourdis, Christos

Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 131-136.

We show that the elliptic problem $\Delta u+f\left(u\right)=0$ in ${ℝ}^N$, $N\ge 1$, with $f\in C^1\left(ℝ\right)$ and $f\left(0\right)=0$ does not have nontrivial stable solutions that decay to zero at infinity, provided that $f$ is nonincreasing near the origin. As a corollary, we can show that any two nontrivial solutions that decay to zero at infinity must intersect each other, provided that at least one of them is sign-changing. This property was previously known only in the case where both solutions are positive with a different approach. We also discuss implications of our main result on the existence of monotone heteroclinic solutions to the corresponding reaction-diffusion equation.

[1] Alberti, Giovanni; Ambrosio, Luigi; Cabré, Xavier On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., Volume 65 (2001) no. 1-3, pp. 9-33 | DOI | MR | Zbl

[2] Berestycki, Henri; Caffarelli, Luis; Nirenberg, Louis Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 25 (1997) no. 1-2, pp. 69-94 | Numdam | MR | Zbl

[3] Berestycki, Henri; Nirenberg, Louis On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat., Nova Sér., Volume 22 (1991) no. 1, pp. 1-37 | DOI | MR | Zbl

[4] Berestycki, Henri; Nirenberg, Louis; Varadhan, S. R. Srinivasa The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Commun. Pure Appl. Math., Volume 47 (1994) no. 1, pp. 47-92 | DOI | MR | Zbl

[5] Busca, Jérôme; Jendoubi, Mohamed Ali; Poláčik, Peter Convergence to equilibrium for semilinear parabolic problems in ${ℝ}^N$, Commun. Partial Differ. Equations, Volume 27 (2002) no. 9-10, pp. 1793-1814 | DOI | MR | Zbl

[6] Cabré, Xavier Uniqueness and stability of saddle-shaped solutions to the Allen–Cahn equation, J. Math. Pures Appl., Volume 98 (2012) no. 3, pp. 239-256 | DOI | MR | Zbl

[7] Cabré, Xavier; Capella, Antonio On the stability of radial solutions of semilinear elliptic equations in all of ${ℝ}^n$, C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 10, pp. 769-774 | DOI | MR | Zbl

[8] Cabré, Xavier; Figalli, Alessio; Ros-Oton, Xavier; Serra, Joaquim Stable solutions to semilinear elliptic equations are smooth up to dimension $9$, Acta Math., Volume 224 (2020) no. 2, pp. 187-252 | DOI | MR

[9] Dancer, Edward Norman Stable and finite Morse index solutions on ${ℝ}^n$ or on bounded domains with small diffusion, Trans. Am. Math. Soc., Volume 357 (2005) no. 3, pp. 1225-1243 | DOI | MR | Zbl

[10] Derrick, G. H. Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys., Volume 5 (1964), pp. 1252-1254 | DOI | MR

[11] Dupaigne, Louis Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 143, CRC Press, 2011 | MR | Zbl

[12] Dupaigne, Louis; Farina, Alberto Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains (2019) (https://arxiv.org/abs/1912.11639)

[13] Farina, Alberto Liouville-Type Theorems for Elliptic Problems, Handbook of Differential Equations 4: Stationary Partial Differential Equations, Elsevier, 2007, pp. 61-116 | Zbl

[14] Gui, Changfeng; Ni, Wei-Ming; Wang, Xuefeng On the stability and instability of positive steady states of a semilinear heat equation in ${ℝ}^n$, Commun. Pure Appl. Math., Volume 45 (1992) no. 9, pp. 1153-1181 | MR | Zbl

[15] Hamel, François; Ninomiya, Hirokazu Localized and expanding entire solutions of reaction-diffusion equations (2020) (https://arxiv.org/abs/2005.07420)

[16] Li, Yi; Ni, Wei-Ming Radial symmetry of positive solutions of nonlinear elliptic equations in ${ℝ}^n$, Commun. Partial Differ. Equations, Volume 18 (1993) no. 5-6, pp. 1043-1054 | DOI | MR | Zbl

[17] Pao, Chia-Ven Nonlinear Parabolic and Elliptic Equations, Plenum Press, 1992 | Zbl

[18] Poláčik, Peter; Yanagida, Eiji On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., Volume 327 (2003) no. 4, pp. 745-771 | MR | Zbl

[19] Quittner, Pavol; Souplet, Philippe Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, 2019 | Zbl

[20] Sattinger, David H. Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, 309, Springer, 1973 | MR | Zbl

[21] Strauss, Walter A. Stable and unstable states of nonlinear wave equations, Nonlinear partial differential equations (Contemporary Mathematics), Volume 17, American Mathematical Society, 1983, pp. 429-441 (Proceedings of a Conference held at the University of New Hampshire, Durham/ New Hampshire, June 20-26, 1982, sponsored by the American Mathematical Society) | DOI | MR

[22] Villegas, Salvador Asymptotic behavior of stable radial solutions of semilinear elliptic equations in ${ℝ}^N$, J. Math. Pures Appl., Volume 88 (2007) no. 3, pp. 241-250 | DOI | MR | Zbl

[23] Villegas, Salvador Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of ${ℝ}^n$, Commun. Pure Appl. Anal., Volume 10 (2011) no. 6, pp. 1817-1821 | DOI | MR | Zbl