Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue

Dimitri Mugnai

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 4, page 508-521
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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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 11.4 (2005): 508-521. <http://eudml.org/doc/245432>.

@article{Mugnai2005,
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; Eigenvalues; estimates},
language = {eng},
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/245432},
volume = {11},
year = {2005},
}

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
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; Eigenvalues; estimates
UR - http://eudml.org/doc/245432
ER -

References

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  1. [1] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973) 349–381. Zbl0273.49063
  2. [2] M. Balabane, J. Dolbeault and H. Ounaies, Nodal solutions for a sublinear elliptic equation. Nonlinear Analysis TMA 52 (2003) 219–237. Zbl1087.35033
  3. [3] A. Bahri and P.L. Lions, Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math. 45 (1992) 1205–1215. Zbl0801.35026
  4. [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. Zbl0946.35023
  5. [5] T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equation. Comm. Partial Differ. Equ. 29 (2004) 25–42. Zbl1140.35410
  6. [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. Zbl1094.35041
  7. [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. Zbl1114.35068
  8. [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. Zbl0686.35043
  9. [9] H. Brezis and T. Kato, Remarks on the Scrödinger operator with singular complex potentials. J. Pure Appl. Math. 33 (1980) 137–151. Zbl0408.35025
  10. [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. Zbl0901.35028
  11. [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. Zbl1042.35036
  12. [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. Zbl0935.35049
  13. [13] L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of p -Laplace equations, 1 l t ; p l t ; 2 , via the moving plane method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998) 689–707. Zbl0930.35070
  14. [14] L. Damascelli and F. Pacella, Monotonicity and symmetry results for p -Laplace equations and applications. Adv. Differential Equations 5 (2000) 1179–1200, Zbl1002.35045
  15. [15] P. Drábek and S.B. Robinson, On the Generalization of the Courant Nodal Domain Theorem. J. Differ. Equ. 181 (2002) 58–71. Zbl1163.35449
  16. [16] M. Grossi, F. Pacella and S.L. Yadava, Symmetry results for perturbed problems and related questions. Topol. Methods Nonlinear Anal. (to appear). Zbl1274.35021
  17. [17] S.J. Li and M. Willem, Applications of local linking to critical point theory. J. Math. Anal. Appl. 189 (1995) 6–32. Zbl0820.58012
  18. [18] J. Moser, A new proof of De Giorgi’s theorem. Comm. Pure Appl. Math. 13 (1960) 457–468. Zbl0111.09301
  19. [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. Zbl1102.35040
  20. [20] F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. J. Funct. Anal. 192 (2002) 271–282 Zbl1014.35032
  21. [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). Zbl0609.58002MR845785
  22. [22] M. Struwe, Variational Methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Springer-Verlag (1990). Zbl0746.49010MR1078018
  23. [23] Z.Q. Wang, On a superlinear elliptic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 43–57. Zbl0733.35043

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