Topological Methods in Nonlinear Analysis

An eigenvalue problem for a quasilinear elliptic field equation on $\mathbb R^n$

V. Benci, A. M. Micheletti, and D. Visetti

Full-text: Open access

Abstract

We study the field equation $$ -\Delta u+V(x)u+\varepsilon^r(-\Delta_pu+W'(u))=\mu u $$ on $\mathbb R^n$, with $\varepsilon$ positive parameter. The function $W$ is singular in a point and so the configurations are characterized by a topological invariant: the topological charge. By a min-max method, for $\varepsilon$ sufficiently small, there exists a finite number of solutions $(\mu(\varepsilon),u(\varepsilon))$ of the eigenvalue problem for any given charge $q\in{\mathbb Z}\setminus\{0\}$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 17, Number 2 (2001), 191-211.

Dates
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1471875817

Mathematical Reviews number (MathSciNet)
MR1868898

Zentralblatt MATH identifier
1109.35369

Citation

Benci, V.; Micheletti, A. M.; Visetti, D. An eigenvalue problem for a quasilinear elliptic field equation on $\mathbb R^n$. Topol. Methods Nonlinear Anal. 17 (2001), no. 2, 191--211. https://projecteuclid.org/euclid.tmna/1471875817


Export citation

References

  • , 297–324 \ref\no5 V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators , J. Math. Anal. Appl., 64 (1978) \ref\no6 V. Benci, D. Fortunato, A. Masiello and L. Pisani, Solitons and the electromagnetic field , Math. Z., 232 (1999), 73–102 \ref\no7 V. Benci, D. Fortunato and L. Pisani, Soliton-like solution of a Lorentz invariant equation in dimension $3$ , Rev. Math. Phys., 10 (1998), 315–344 \ref\no8 V. Benci, A. M. Micheletti and D. Visetti, An eigenvalue problem for a quasilinear elliptic field equation
  • , J. Differential Equations, to appear \ref\no9 F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Kluwer Academic Publishers (1991) \ref\no10 P. Bolle, On the Bolza problem , J. Differential Equations, 152 (1999), 274–288 \ref\no11 P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems , Manuscripta Math., 101 (2000)
  • , 325–350 \ref\no12 R. Courant and D. Hilbert, Methods of Mathematical Physics, I , Interscience, New York \ref\no13 C. H. Derrick, Comments on nonlinear wave equations as model for elementary particles , J. Math. Phys., 5 (1964), 1252–1254 \ref\no14 N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge University Press (1993) \ref\no15 T. Kato, Perturbation Theory for Linear Operators, Springer–Verlag, Berlin (1980) \ref\no16 J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod-Gauthier Villar, Paris (1969) \ref\no17 A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine , Boll. Un. Mat. Ital. A (7), 4 (1973), 285–301 \ref\no18 R. S. Palais, Lusternik–Schnirelman theory on Banach manifolds , Topology, 5 (1966)