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.
Mots-clés : 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},
pages = {508--521},
publisher = {EDP-Sciences},
volume = {11},
number = {4},
year = {2005},
doi = {10.1051/cocv:2005017},
mrnumber = {2167872},
zbl = {1103.35032},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2005017/}
}
TY - JOUR AU - Mugnai, Dimitri TI - Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2005 SP - 508 EP - 521 VL - 11 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2005017/ DO - 10.1051/cocv:2005017 LA - en ID - COCV_2005__11_4_508_0 ER -
%0 Journal Article %A Mugnai, Dimitri %T Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue %J ESAIM: Control, Optimisation and Calculus of Variations %D 2005 %P 508-521 %V 11 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2005017/ %R 10.1051/cocv:2005017 %G en %F 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, Tome 11 (2005) no. 4, pp. 508-521. doi : 10.1051/cocv:2005017. http://archive.numdam.org/articles/10.1051/cocv:2005017/
[1] and, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973) 349-381. | Zbl
[2] , and, Nodal solutions for a sublinear elliptic equation. Nonlinear Analysis TMA 52 (2003) 219-237. | Zbl
[3] and, Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math. 45 (1992) 1205-1215. | Zbl
[4] , and, On the Morse indices of sign changing solutions of nonlinear elliptic problems. Math. Z. 233 (2000) 655-677. | Zbl
[5] , and, Sign changing solutions of superlinear Schrödinger equation. Comm. Partial Differ. Equ. 29 (2004) 25-42. | Zbl
[6] and, A note on additional properties of sign changing solutions to superlinear elliptic equations. Topol. Methods Nonlinear Anal. 22 (2003) 1-14. | Zbl
[7] and, Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 259-281. | EuDML | Numdam | Zbl
[8] and, 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
[9] and, Remarks on the Scrödinger operator with singular complex potentials. J. Pure Appl. Math. 33 (1980) 137-151. | Zbl
[10] , and, 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
[11] , On the nodal set of the second eigenfunction of the Laplacian in symmetric domains in . Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000) 175-181. | Zbl
[12] , and, 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
[13] and, Monotonicity and symmetry of solutions of -Laplace equations, , via the moving plane method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998) 689-707. | Numdam | Zbl
[14] and, Monotonicity and symmetry results for -Laplace equations and applications. Adv. Differential Equations 5 (2000) 1179-1200, | Zbl
[15] and, On the Generalization of the Courant Nodal Domain Theorem. J. Differ. Equ. 181 (2002) 58-71.
[16] , and, Symmetry results for perturbed problems and related questions. Topol. Methods Nonlinear Anal. (to appear).
[17] and, Applications of local linking to critical point theory. J. Math. Anal. Appl. 189 (1995) 6-32. | Zbl
[18] , A new proof of De Giorgi's theorem. Comm. Pure Appl. Math. 13 (1960) 457-468. | Zbl
[19] , 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
[20] , Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. J. Funct. Anal. 192 (2002) 271-282 | Zbl
[21] , 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 | Zbl
[22] , Variational Methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Springer-Verlag (1990). | MR | Zbl
[23] , On a superlinear elliptic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 43-57. | Numdam | Zbl
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