Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T23:29:45.731Z Has data issue: false hasContentIssue false

Asymptotic behaviour, nodal lines and symmetry properties forsolutions of superlinear elliptic equations nearan eigenvalue

Published online by Cambridge University Press:  15 September 2005

Dimitri Mugnai*
Affiliation:
Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli 1, 06123 Perugia, Italy; [email protected]
Get access

Abstract

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

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ambrosetti, A. and Rabinowitz, P.H., Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973) 349381. CrossRef
Balabane, M., Dolbeault, J. and Ounaies, H., Nodal solutions for a sublinear elliptic equation. Nonlinear Analysis TMA 52 (2003) 219237. CrossRef
Bahri, A. and Lions, P.L., Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math. 45 (1992) 12051215. CrossRef
Bartsch, T., Chang, K.C. and Wang, Z.Q., On the Morse indices of sign changing solutions of nonlinear elliptic problems. Math. Z. 233 (2000) 655677. CrossRef
Bartsch, T., Liu, Z. and Weth, T., Sign changing solutions of superlinear Schrödinger equation. Comm. Partial Differ. Equ. 29 (2004) 2542. CrossRef
Bartsch, T. and Weth, T., A note on additional properties of sign changing solutions to superlinear elliptic equations. Topol. Methods Nonlinear Anal. 22 (2003) 114. CrossRef
Bartsch, T. and Weth, T., Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 259281. CrossRef
Benci, V. and Fortunato, D., A remark on the nodal regions of the solutions of some superlinear elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 123128. CrossRef
Brezis, H. and Kato, T., Remarks on the Scrödinger operator with singular complex potentials. J. Pure Appl. Math. 33 (1980) 137151.
Castro, A., Cossio, J. and Neuberger, J.M., 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.
Damascelli, L., On the nodal set of the second eigenfunction of the Laplacian in symmetric domains in $\mathbb{R}^N$ . Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000) 175181.
Damascelli, L., Grossi, M. and Pacella, F., 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) 631652. CrossRef
Damascelli, L. and Pacella, F., Monotonicity and symmetry of solutions of p-Laplace equations, $1 < p < 2$ , via the moving plane method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998) 689707.
Damascelli, L. and Pacella, F., Monotonicity and symmetry results for p-Laplace equations and applications. Adv. Differential Equations 5 (2000) 11791200,
Drábek, P. and Robinson, S.B., On the Generalization of the Courant Nodal Domain Theorem. J. Differ. Equ. 181 (2002) 5871. CrossRef
M. Grossi, F. Pacella and S.L. Yadava, Symmetry results for perturbed problems and related questions. Topol. Methods Nonlinear Anal. (to appear).
Li, S.J. and Willem, M., Applications of local linking to critical point theory. J. Math. Anal. Appl. 189 (1995) 632. CrossRef
Moser, J., A new proof of De Giorgi's theorem. Comm. Pure Appl. Math. 13 (1960) 457468. CrossRef
Mugnai, D., Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem. Nonlinear Differ. Equ. Appl. 11 (2004) 379391. CrossRef
Pacella, F., Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. J. Funct. Anal. 192 (2002) 271282 CrossRef
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).
Wang, Z.Q., On a superlinear elliptic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 4357. CrossRef