Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue
ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 4, p. 508-521

We give the precise behaviour of some solutions of a nonlinear elliptic B.V.P. in a bounded domain when a parameter approaches an eigenvalue of the principal part. If the nonlinearity has some regularity and the domain is for example convex, we also prove a nonlinear version of Courant's Nodal theorem.

DOI : https://doi.org/10.1051/cocv:2005017
Classification:  35B40,  35J65
Keywords: eigenvalues, ${L}^{\infty }-{H}_{0}^{1}$ estimate, nodal lines, symmetries
@article{COCV_2005__11_4_508_0,
author = {Mugnai, Dimitri},
title = {Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {11},
number = {4},
year = {2005},
pages = {508-521},
doi = {10.1051/cocv:2005017},
zbl = {1103.35032},
mrnumber = {2167872},
language = {en},
url = {http://www.numdam.org/item/COCV_2005__11_4_508_0}
}

Mugnai, Dimitri. Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 4, pp. 508-521. doi : 10.1051/cocv:2005017. http://www.numdam.org/item/COCV_2005__11_4_508_0/

[1] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973) 349-381. | Zbl 0273.49063

[2] M. Balabane, J. Dolbeault and H. Ounaies, Nodal solutions for a sublinear elliptic equation. Nonlinear Analysis TMA 52 (2003) 219-237. | Zbl 1087.35033

[3] A. Bahri and P.L. Lions, Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math. 45 (1992) 1205-1215. | Zbl 0801.35026

[4] T. Bartsch, K.C. Chang and Z.Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems. Math. Z. 233 (2000) 655-677. | Zbl 0946.35023

[5] T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equation. Comm. Partial Differ. Equ. 29 (2004) 25-42. | Zbl 1140.35410

[6] T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations. Topol. Methods Nonlinear Anal. 22 (2003) 1-14. | Zbl 1094.35041

[7] T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 259-281. | Numdam | Zbl 1114.35068

[8] V. Benci and D. Fortunato, A remark on the nodal regions of the solutions of some superlinear elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 123-128. | Zbl 0686.35043

[9] H. Brezis and T. Kato, Remarks on the Scrödinger operator with singular complex potentials. J. Pure Appl. Math. 33 (1980) 137-151. | Zbl 0408.35025

[10] A. Castro, J. Cossio and J.M. Neuberger, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems. Electron. J. Differ. Equ. 2 (1998) 18. | Zbl 0901.35028

[11] L. Damascelli, On the nodal set of the second eigenfunction of the Laplacian in symmetric domains in ${ℝ}^{N}$. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000) 175-181. | Zbl 1042.35036

[12] L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle. Ann. Inst. H. Poincaré. Anal. Non Linéaire 16 (1999) 631-652. | Numdam | Zbl 0935.35049

[13] L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of $p$-Laplace equations, $1, via the moving plane method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998) 689-707. | Numdam | Zbl 0930.35070

[14] L. Damascelli and F. Pacella, Monotonicity and symmetry results for $p$-Laplace equations and applications. Adv. Differential Equations 5 (2000) 1179-1200, | Zbl 1002.35045

[15] P. Drábek and S.B. Robinson, On the Generalization of the Courant Nodal Domain Theorem. J. Differ. Equ. 181 (2002) 58-71. | Zbl pre01764708

[16] M. Grossi, F. Pacella and S.L. Yadava, Symmetry results for perturbed problems and related questions. Topol. Methods Nonlinear Anal. (to appear). | Zbl pre01997215

[17] S.J. Li and M. Willem, Applications of local linking to critical point theory. J. Math. Anal. Appl. 189 (1995) 6-32. | Zbl 0820.58012

[18] J. Moser, A new proof of De Giorgi's theorem. Comm. Pure Appl. Math. 13 (1960) 457-468. | Zbl 0111.09301

[19] D. Mugnai, Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem. Nonlinear Differ. Equ. Appl. 11 (2004) 379-391. | Zbl 1102.35040

[20] F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. J. Funct. Anal. 192 (2002) 271-282 | Zbl 1014.35032

[21] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, RI (1986). | MR 845785 | Zbl 0609.58002

[22] M. Struwe, Variational Methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Springer-Verlag (1990). | MR 1078018 | Zbl 0746.49010

[23] Z.Q. Wang, On a superlinear elliptic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 43-57. | Numdam | Zbl 0733.35043