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Infinitely many solutions of a symmetric Dirichlet problem. (English) Zbl 0799.35071
We seek solutions $u=(u\sb 1,\dots, u\sb m): \overline{\Omega} \to \bbfR\sp m$ of the nonlinear Dirichlet problem $$\Delta u+F\sb u(u)=u \quad \text{in }\Omega, \qquad u=0 \quad \text{on } \partial\Omega. \tag D$$ Here $\Omega$ is a bounded domain in $\bbfR\sp n$ with smooth boundary and $F: \bbfR\sp m \to\bbfR$ is $C\sp 1$ and satisfies certain growth conditions. We investigate which kind of symmetries guarantee the existence of infinitely many solutions of (D).

35J65Nonlinear boundary value problems for linear elliptic equations
58J70Invariance and symmetry properties
35J20Second order elliptic equations, variational methods
Full Text: DOI
[1] Ambrosetti, A.; Rabinowitz, P. H.: Dual variational methods in critical point theory and applications. J. funct. Analysis 14, 349-381 (1973) · Zbl 0273.49063
[2] Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations. CBMS regional conference series in mathematics. 65 (1986)
[3] Michalek, R.: A zp borsuk-Ulam theorem and index theory with a multiplicity result in partial differential equations. Nonlinear analysis 13, 957-968 (1989) · Zbl 0688.58040
[4] Krasnoselski, M. A.: On special coverings of a finite-dimensional sphere. Dokl. akad. Nauk. SSSR 103, 961-964 (1955)
[5] Krasnoselski, M. A.: Topological methods in the theory of nonlinear integral equations. (1964)
[6] Bröcker, T.; Dieck, T. Tom: Representations of compact Lie groups. Graduate texts in mathematics 98 (1985) · Zbl 0581.22009
[7] Friedmann, A.: Partial differential equations. (1969)
[8] Benci, V.: A geometrical index for the group S1 and some applications to the study of periodic solutions of ordinary differential equations. Communs pure appl. Math. 34, 393-432 (1981) · Zbl 0447.34040
[9] Bartsch, T.: On the genus of representation spheres. Comment. math. Helv. 65, 85-95 (1990) · Zbl 0704.57024
[10] Clapp, M.; Puppe, D.: Critical point theory with symmetrics. J. reine angew. Math. 418, 1-29 (1991) · Zbl 0722.58011
[11] Bartsch, T.; Clapp, M.; Puppe, D.: A mountain pass theorem for actions of compact Lie groups. J. reine angew. Math. 419, 55-56 (1991) · Zbl 0731.58016
[12] Bartsch T., A simple proof of the degree formula for Z/p-equivariant maps, Math. 2 (to appear). · Zbl 0790.55003
[13] Steinlein, H.: Borsuk’s antipodal theorem and its generalizations and applications: a survey. Méth. topol. En anal. Non linéaire, comp. Rend. coll. Par A. Granas, 166-235 (1985)
[14] Bartsch, T.: On the existence of borsuk-Ulam theorems. Topology 31, 533-543 (1992) · Zbl 0770.55003
[15] Ihrig, E.; Golubitsky, M.: Pattern selection with $O(3)$-symmetry. Physica 13D, 1-33 (1984) · Zbl 0581.22021
[16] Lauterbach, R.: Bifurcation with $O(3)$-symmetry. Habilitation thesis (1988) · Zbl 0659.58007