Lane–Emden problems: Asymptotic behavior of low energy nodal solutions

Grossi, Massimo; Grumiau, Christopher; Pacella, Filomena

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 1, pp. 121-140.

We study the nodal solutions of the Lane–Emden–Dirichlet problem

$$\{\begin{array}{cc}-\Delta u={\left|u\right|}^{p-1}u,\hfill & \text{in}\Omega ,\hfill \\ u=0,\hfill & \text{on}\partial \Omega ,\hfill \end{array}$$

where Ω is a smooth bounded domain in ${ℝ}^2$ and $p>1$. We consider solutions $u_p$ satisfying

$$p\underset{\Omega }{\int }{|\nabla u_p|}^2\to 16\pi e\text{as}p\to +\infty$$

and we are interested in the shape and the asymptotic behavior as $p\to +\infty$.First we prove that (⁎) holds for least energy nodal solutions. Then we obtain some estimates and the asymptotic profile of this kind of solutions. Finally, in some cases, we prove that $p{u}_p$ can be characterized as the difference of two Greenʼs functions and the nodal line intersects the boundary of Ω, for large p.

[1] Adimurthi, Massimo Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc. 132 no. 4 (2004), 1013-1019 | MR | Zbl

[2] Amandine Aftalion, Filomena Pacella, Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains, C. R. Math. Acad. Sci. Paris 339 no. 5 (2004), 339-344 | MR | Zbl

[3] Thomas Bartsch, Tobias Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal. 22 no. 1 (2003), 1-14 | MR | Zbl

[4] Thomas Bartsch, Tobias Weth, Michel Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math. 96 (2005), 1-18 | MR | Zbl

[5] Mohamed Ben Ayed, Khalil El Mehdi, Filomena Pacella, Blow-up and nonexistence of sign changing solutions to the Brezis–Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 no. 4 (2006), 567-589 | EuDML | Numdam | Zbl

[6] Mohamed Ben Ayed, Khalil El Mehdi, Filomena Pacella, Blow-up and nonexistence of sign changing solutions to the Brezis–Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 no. 4 (2006), 567-589 | EuDML | Numdam | MR | Zbl

[7] Mohamed Ben Ayed, Khalil El Mehdi, Filomena Pacella, Classification of low energy sign-changing solutions of an almost critical problem, J. Funct. Anal. 250 no. 2 (2007), 347-373 | MR | Zbl

[8] Denis Bonheure, Vincent Bouchez, Christopher Grumiau, Jean Van Schaftingen, Asymptotics and symmetries of least energy nodal solutions of Lane–Emden problems with slow growth, Commun. Contemp. Math. 10 no. 4 (2008), 609-631 | MR | Zbl

[9] Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. Théorie et applications, Masson, Paris (1983) | MR | Zbl

[10] Alfonso Castro, Jorge Cossio, John M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 no. 4 (1997), 1041-1053 | MR | Zbl

[11] Khalil El Mehdi, Massimo Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Adv. Nonlinear Stud. 4 no. 1 (2004), 15-36 | MR | Zbl

[12] Pierpaolo Esposito, Monica Musso, Angela Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differential Equations 227 no. 1 (2006), 29-68 | MR | Zbl

[13] Pierpaolo Esposito, Monica Musso, Angela Pistoia, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, Proc. Lond. Math. Soc. (3) 94 no. 2 (2007), 497-519 | MR | Zbl

[14] Basilis Gidas, Wei Ming Ni, Louis Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 no. 3 (1979), 209-243 | MR | Zbl

[15] Christopher Grumiau, Christophe Troestler, Oddness of least energy nodal solutions on radial domains, Electron. J. Differ. Equ. Conf. 18 (2010), 23-31 | EuDML | MR | Zbl

[16] Zheng-Chao Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 no. 2 (1991), 159-174 | EuDML | Numdam | MR | Zbl

[17] Li Ma, Juncheng C. Wei, Convergence for a Liouville equation, Comment. Math. Helv. 76 no. 3 (2001), 506-514 | MR | Zbl

[18] Antonios D. Melas, On the nodal line of the second eigenfunction of the Laplacian in ${𝐑}^2$, J. Differential Geom. 35 no. 1 (1992), 255-263 | MR | Zbl

[19] Filomena Pacella, Tobias Weth, Symmetry of solutions to semilinear elliptic equations via Morse index, Proc. Amer. Math. Soc. 135 no. 6 (2007), 1753-1762 | MR | Zbl

[20] Xiaofeng Ren, Juncheng Wei, Single-point condensation and least-energy solutions, Proc. Amer. Math. Soc. 124 no. 1 (1996), 111-120 | MR | Zbl

[21] Xiaofeng Ren, Juncheng Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc. 343 no. 2 (1994), 749-763 | MR | Zbl