Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue
ESAIM: Control, Optimisation and Calculus of Variations (2010)
- Volume: 11, Issue: 4, page 508-521
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topMugnai, Dimitri. "Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue." ESAIM: Control, Optimisation and Calculus of Variations 11.4 (2010): 508-521. <http://eudml.org/doc/90775>.
@article{Mugnai2010,
abstract = {
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.
},
author = {Mugnai, Dimitri},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Eigenvalues; $L^\infty-H_0^1$ estimate; nodal lines;
symmetries.; estimates; symmetries},
language = {eng},
month = {3},
number = {4},
pages = {508-521},
publisher = {EDP Sciences},
title = {Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue},
url = {http://eudml.org/doc/90775},
volume = {11},
year = {2010},
}
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
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 4
SP - 508
EP - 521
AB -
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.
LA - eng
KW - Eigenvalues; $L^\infty-H_0^1$ estimate; nodal lines;
symmetries.; estimates; symmetries
UR - http://eudml.org/doc/90775
ER -
References
top- A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal.14 (1973) 349–381.
- M. Balabane, J. Dolbeault and H. Ounaies, Nodal solutions for a sublinear elliptic equation. Nonlinear Analysis TMA52 (2003) 219–237.
- A. Bahri and P.L. Lions, Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math.45 (1992) 1205–1215.
- 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.
- T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equation. Comm. Partial Differ. Equ.29 (2004) 25–42.
- 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.
- T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann. Inst. H. Poincaré Anal. Non Linéaire22 (2005) 259–281.
- V. Benci and D. Fortunato, A remark on the nodal regions of the solutions of some superlinear elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A111 (1989) 123–128.
- H. Brezis and T. Kato, Remarks on the Scrödinger operator with singular complex potentials. J. Pure Appl. Math.33 (1980) 137–151.
- 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.
- L. Damascelli, 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.
- 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 Linéaire16 (1999) 631–652.
- L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of p-Laplace equations, , via the moving plane method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)26 (1998) 689–707.
- L. Damascelli and F. Pacella, Monotonicity and symmetry results for p-Laplace equations and applications. Adv. Differential Equations5 (2000) 1179–1200,
- P. Drábek and S.B. Robinson, On the Generalization of the Courant Nodal Domain Theorem. J. Differ. Equ.181 (2002) 58–71.
- M. Grossi, F. Pacella and S.L. Yadava, Symmetry results for perturbed problems and related questions. Topol. Methods Nonlinear Anal. (to appear).
- S.J. Li and M. Willem, Applications of local linking to critical point theory. J. Math. Anal. Appl.189 (1995) 6–32.
- J. Moser, A new proof of De Giorgi's theorem. Comm. Pure Appl. Math.13 (1960) 457–468.
- 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.
- F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. J. Funct. Anal.192 (2002) 271–282
- 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).
- M. Struwe, Variational Methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Springer-Verlag (1990).
- Z.Q. Wang, On a superlinear elliptic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire8 (1991) 43–57.
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.