## Infinitely many solutions of a symmetric Dirichlet problem.(English)Zbl 0799.35071

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

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 58J70 Invariance and symmetry properties for PDEs on manifolds 35J20 Variational methods for second-order elliptic equations
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### References:

 [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, Vol. 65 (1986), American Mathematical Society: American Mathematical Society Providence, Rhode Island [3] Michalek, R., A $$Z^p$$ 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), (In Russian.) [5] Krasnoselski, M. A., Topological Methods in the Theory of Nonlinear Integral Equations (1964), Macmillan: Macmillan New York [6] Bröcker, T.; Tom Dieck, T., Representations of compact Lie groups, (Graduate Texts in Mathematics, Vol. 98 (1985), Springer: Springer Berlin) · Zbl 0581.22009 [7] Friedmann, A., Partial Differential Equations (1969), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York [8] Benci, V., A geometrical index for the group $$S^1$$ 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 [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 (1985), Presses de l’Univ. de Monteal), 166-235 · Zbl 0573.55003 [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), University of Augsburg) · Zbl 0722.58031
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